Prediction of pressure drop from multivariate polynomial regression for membrane-based air-to-air energy recovery ventilator cores

Abstract

Ventilation is an active control method that can improve the indoor environment by removing indoor air pollution. The membrane-based air-to-air energy recovery ventilator is a ventilation device that utilizes the heat exchange between the discharged polluted air and fresh outdoor air during ventilation. This study analyzed the pressure drop characteristics of the ventilator core and aimed to develop an equation to predict the pressure drop using polynomial regression based on the size of the core, the airflow rate, and the opening ratio of the core. Within elements having the same opening ratio, the channel velocity (u_c) and equivalent pressure drop (P_e) values showed a linear relationship, regardless of the element size. The opening ratio of the elements varied depending on the cell height. To analyze the correlation between the opening ratio of the element, substantial flow rate inside the element, and equivalent pressure drop, a multivariate polynomial regression analysis was conducted for the three variables. The difference between the values of the derived prediction formula and the experimental value was found to be within 5% across the entire range in which the experiment was conducted. Using the interaction formula of equivalent pressure drop, channel velocity, and opening ratio derived in this study, the pressure drop across the entire driving area could be predicted for an arbitrary cell height and width of the ventilator core, as well as the operating airflow rate.

Introduction

Recently, human life has changed from agricultural to industrialized-oriented societies. Along with rapid changes in society, the space where people spend most of their time during the day has shifted from being outside to being inside. Owing to these changes, humans spend more than 80% of their time in indoor spaces, such as offices, houses, and schools, and many efforts have been made to make these environments more pleasant (Lim et al. 2012).

Ventilation is an active control method that can make indoor environments more pleasant by removing indoor air pollution (Bearg 1993; Cho, Oh, and Ahn 2017; Huizing, Chen, and Wong 2015). The trend of constructing high-rise buildings, such as apartments and offices, with increased airtightness has increased the importance of ventilation equipment because significant amounts of harmful chemicals accumulate indoors (Daisey, Angell, and Apte 2003; Jan, Curtiss, and Rabl 2002). As ventilation replaces polluted indoor air by bringing in fresh outdoor air, the heating and cooling loads of a building significantly increase in summer and winter when the indoor–outdoor temperature difference is large (Drost 1993; Fisk and Turiel 1983). Such problems have generated the need for an eco-friendly ventilation system that can secure a pleasant indoor air environment while reducing energy consumption by reducing heating and cooling loads (Kalbasi, Ruhani, and Rostami 2019; Ke and Yanming 2009).

The membrane-based air-to-air energy recovery ventilator (MERV) is a ventilation device that utilizes the heat exchange between discharged polluted air and fresh outdoor air during ventilation (Connor et al. 2016; Swiercz 2021; Zhang et al. 2000). It recovers and recycles the heat energy generated through the heating and cooling of a building, saving energy while simultaneously purifying and supplying outdoor air, making it an effective ventilation device that can supply fresh air indoors using a simple system (Lekshminarayanan, Croal, and Maisonneuve 2020; Mansur et al. 2021; Souifi et al., 2021). The core part of an MERV is the enthalpy exchanger, where the heat exchange takes place, which comprises a membrane and spacer, as shown in Figure 1.

Fig. 1. A schematic of an element core.

Heat exchange occurs via the transfer of temperature and moisture between the internal and external air through a liner. The spacer forms an airflow path and maintains the overall phase of the enthalpy exchanger (Armatis and Fronk 2017; Busto et al. 2017; Masitah, Ahmad, and Yatim 2015). The paper used in the liner of an enthalpy exchanger is often coated with a moisture absorbent to increase its absorption ability. Additionally, to prevent the mixing of the inflow and outflow air, the fiber spacing is often made denser by pressing the paper through a cylindrical roller (calendering). However, because the main purpose of a spacer is to maintain the airflow path and shape of the element, papers that have not undergone such special processing were used for these experiments (Kim 2016; Kwak et al. 2007).

Significant research has been conducted on MERV performance, with recent studies seeking to predict the performance of the ventilator core through experimentation, numerical and effective number of transfer units (NTU) methods, and computational analysis (computational fluid dynamics [CFD]). The numerical and effective NTU method can reflect the variables for shape and material more easily than other methods; therefore, it is widely used for the analysis of variables that are difficult to implement experimentally (Min and Duan 2016; Min and Su 2010; Zhang 2016). Experimental methods are most frequently used because they can analyze both energy efficiency and differential pressure (Engarnevis et al. 2018; Gao et al. 2021; Kwak and Bai 2009; Papakostas and Kiosis 2015; Siegele 2019; Zhang et al. 2017). These methods can be combined to analyze the energy efficiency performance of the MERV, and a comparative analysis of their results can be performed (Kim and Kim 2018; Lee et al. 2012; Qiu et al. 2019; Siegele and Ochs 2019; Zhang 2012). Computational fluid dynamics is used less frequently than other methods; however, it can be useful when the flow rate and temperature distribution inside an element are important (Waked et al. 2015; Waked et al. 2013; Xiana, Tang, and Mo 2018). Energy efficiency performance can also be analyzed using polynomial regression (Gao et al. 2021; Qiu et al. 2019).

While many studies analyzing the energy efficiency performance of MERV have been conducted, few studies have attempted to predict the pressure drop of the ventilator core, which is essential for the design of a blower fan, the MERV component that consumes the most energy. No existing studies have focused on predicting the pressure drop of a core based on the size and the opening ratio of the core and the airflow rate. This study analyzed the pressure drop characteristics of the ventilator core through experimentation and, using polynomial regression, aimed to develop a pressure drop prediction equation based on the size and the opening ratio of the core and airflow rate.

Methodology

The pressure drop was measured under the condition of changing the airflow rate for the nine ventilator cores, as shown in Table 1.

Table 1. The test-core dimensions.

Core No.	H (mm)	W (mm)	L (mm)	Hc (mm)	Wc (mm)	Dh (mm)
1	150	150	150	1.8	3.6	1.49
2	150	150	150	2.0	4.0	1.66
3	150	150	150	2.6	5.2	2.15
4	250	250	250	1.8	3.6	1.49
5	250	250	250	2.0	4.0	1.66
6	250	250	250	2.6	5.2	2.15
7	350	350	350	1.8	3.6	1.49
8	350	350	350	2.0	4.0	1.66
9	350	350	350	2.6	5.2	2.15

The thicknesses of the membrane and spacer used in this experiment are listed in Table 2.

Table 2. Membrane and spacer specifications.

	Membrane	Spacer
Thickness (mm)	0.0406	0.0732
Material	cellulose	cellulose

The actual flow velocity inside the element was calculated to analyze the pressure drop and predict the pressure drop under conditions such as the size and airflow rate of an arbitrary element. The flow velocity at the front end of the core element of the enthalpy exchanger is presented in Equation 1(1) $$u=\frac{V}{H\times L}$$(1) as a value obtained by dividing the cross-sectional area of the ventilator core by the measured airflow rate (V). The ventilator core comprised the intersection of a membrane and spacer, with high- and low-temperature sides each occupying 50% of the flow space and blocking 50% of the front part of the ventilator cores. As a portion of the remaining 50% of the space was occupied by the spacer and membrane, the actual cross-sectional flow area was less than 50% of the front surface of the ventilator core. Equation 2(2) $$R_{fa}=0.5\times \left[1-\left(\frac{W_c{\times \mathrm{T}}_m+2\times \sqrt{H_c\times H_c+0.5W_c\times 0.5W_c}\times T_s}{H_c\times W_c}\right)\right]$$(2) presents the ratio of the actual flow space to the front surface of the ventilator core obtained from the height, width, and thickness of the element defined in Tables 1 and 2. Through the flow rate at the front end of the ventilator core and the opening ratio, the channel velocity within the element was obtained, as shown in Equation 3(3) $$u_c=u\frac{1}{R_{fa}}$$(3) . (1) $$u=\frac{V}{H\times L}$$(1) (2) $$R_{fa}=0.5\times \left[1-\left(\frac{W_c{\times \mathrm{T}}_m+2\times \sqrt{H_c\times H_c+0.5W_c\times 0.5W_c}\times T_s}{H_c\times W_c}\right)\right]$$(2) (3) $$u_c=u\frac{1}{R_{fa}}$$(3)

Figure 2 shows the ventilator cores used in this study.

Fig. 2. The ventilator cores used in the experiment under three size conditions. (L = 150, 250, and 350 mm).

The pressure drop was measured in the flow rate range of 50–549 cubic meters per hour for the nine ventilator cores. The experimental volumetric flow rate, flow velocity, channel velocity, and Reynolds number for each ventilator core are presented in Table 3.

Table 3. Test conditions.

(a) Volumetric flow rate and flow velocity

Core no.	Volumetric flow rate of test (m^3/h)	Flow velocity(m/s)
1	58, 98, 149, 200, 249	0.72, 1.21, 1.84, 2.47, 3.07
2	100, 149, 201, 252, 297	1.23, 1.84, 2.48, 3.11, 3.67
3	100, 149, 199, 249, 298, 344	1.23, 1.84, 2.46, 3.07, 3.68, 4.25
4	106, 151, 199, 251, 298, 348, 400	0.47, 0.67, 0.88, 1.12, 1.32, 1.55, 1.78
5	98, 149, 196, 249, 298, 348, 400	0.44, 0.66, 0.87, 1.11, 1.32, 1.55, 1.78
6	152, 198, 251, 301, 351, 402, 450	0.68, 0.88, 1.12, 1.34, 1.56, 1.79, 2.00
7	148, 196, 248, 300, 351, 397, 450, 501	0.34, 0.44, 0.56, 0.68, 0.80, 0.90, 1.02, 1.14
8	157, 195, 249, 298, 350, 400, 450, 499	0.36, 0.44, 0.56, 0.68, 0.79, 0.91, 1.02, 1.13
9	155, 198, 253, 300, 347, 397, 449, 502, 549	0.35, 0.45, 0.57, 0.68, 0.79, 0.90, 1.02, 1.14, 1.24

(b) Channel velocity and Reynolds number.

Core no.	Channel velocity (m/s)	Reynolds number
1	1.56, 2.63, 4.00, 5.37, 6.68	149, 251, 382, 512, 638
2	2.66, 3.96, 5.35, 6.71, 7.90	282, 421, 567, 711, 838
3	2.61, 3.89, 5.20, 6.51, 7.79, 8.99	360, 537, 717, 898, 1 074, 1 240
4	1.02, 1.46, 1.92, 2.43, 2.88, 3.36, 3.87	98, 139, 184, 232, 275, 321, 369
5	0.94, 1.43, 1.88, 2.39, 2.85, 3.33, 3.83	100, 151, 199, 253, 303, 354, 406
6	1.43, 1.86, 2.36, 2.83, 3.30, 3.78, 4.23	197, 257, 326, 391, 455, 522, 584
7	0.73, 0.97, 1.22, 1.48, 1.73, 1.96, 2.22, 2.47	70, 92, 117, 141, 165, 187, 212, 236
8	0.77, 0.95, 1.22, 1.46, 1.71, 1.95, 2.20, 2.44	81, 101, 129, 154, 181, 207, 233, 259
9	0.74, 0.95, 1.21, 1.44, 1.67, 1.91, 2.16, 2.41, 2.64	103, 131, 168, 199, 230, 263, 297, 332, 363

The airflow rate was measured using a chamber-type multi-nozzle fan tester, as shown in Figure 3. The differential pressure before and after the flow nozzle was used to measure the airflow rate using the flow nozzle. A differential pressure transmitter (EJA110A) with an accuracy of ± 0.065% was used to measure the differential pressure.

Fig. 3. The multi-nozzle fan tester.

Figure 4 shows the connection of each ventilator core used for the pressure drop experiment, where the airflow rate was measured in a stabilized state and the pressure drop was measured at the front and rear ends of the ventilator cores.

Fig. 4. Pressure drop test setup schematic.

To measure the pressure drop, a differential pressure sensor (Beck 984 A.543) was connected to a data logger (KEASIGHT DAQ970A) to enable real-time measurement. The differential pressure sensor has an accuracy of ±1%. According to Gao and Qiu’s studies (Gao et al. 2021; Qiu et al. 2019), the expanded measurement uncertainties of P (pressure drop) and V (airflow rate) can be calculated using Equations 4(4) $$U\left(\mathrm{P}\right)=K\times \sqrt{{\sum }_{i=1}^5\left[\right(\frac{\partial P}{\partial x_i}{)}^2{u}_c(x_i{)}^2]}$$(4) and 5(5) $$U\left(V\right)=K\times \sqrt{{\sum }_{i=1}^5\left[\right(\frac{\partial V}{\partial x_i}{)}^2{u}_c(x_i{)}^2]}$$(5) . In this study, K (coverage factor) was defined as 1.96, which obtains a 95% confidence level (Farrance and Frenkel 2012). (4) $$U\left(\mathrm{P}\right)=K\times \sqrt{{\sum }_{i=1}^5\left[\right(\frac{\partial P}{\partial x_i}{)}^2{u}_c(x_i{)}^2]}$$(4) (5) $$U\left(V\right)=K\times \sqrt{{\sum }_{i=1}^5\left[\right(\frac{\partial V}{\partial x_i}{)}^2{u}_c(x_i{)}^2]}$$(5)

In this study, the maximum expanded measurement uncertainty of the Reynolds Number was ±0.13%. Expanded measurement uncertainties of pressure drop and friction factor are illustrated in Figures 5 and 8.

Fig. 5. Pressure drop by airflow rate change for each ventilator core.

Case study and results

The effects of core and channel sizes

Figure 5 shows a plot of the pressure drops measured under the experimental conditions listed in Table 3.

The pressure drop also increased with an increased airflow rate. For H_c = 1.8 and 2.0 mm, the differences between the values were small. For H_c = 2.6 mm, with a large frontal area ratio, the pressure drop at the same airflow rate was smaller than that at 1.8 and 2.0 mm.

Figure 6 shows a set of plots summarizing the results by ventilator core size under the same conditions (H_c = 1.8, 2.0, and 2.6 mm). As the size of the ventilator core increased, the pressure drop decreased; when H_c increased, the pressure drop also decreased.

Fig. 6. Pressure drop by airflow rate change for each ventilator core with the same H_c.

The effects of channel velocity

Figure 7 shows the results of redefining the x-axis as the Reynolds number in the ventilator core, as obtained from Equation 8(8) $$Re=\frac{u_c{D}_h{\rho }_{\mathit{air}}}{{\mu }_{\mathit{air}}}$$(8) , based on the results shown in Figure 5. This shows that there was a significantly greater pressure drop when the ventilator core size (L) was increased under the same channel velocity conditions within the element.

Fig. 7. Pressure drop by Reynolds number change for each ventilator core.

The smaller the H_c of the ventilator core, the larger the pressure drop. Assuming a steady-state and incompressible flow in each channel of the ventilator core, the pressure drop in the inner channel of the element can be expressed by the Darcy–Weisbach equation, that is, Equation 4(4) $$U\left(\mathrm{P}\right)=K\times \sqrt{{\sum }_{i=1}^5\left[\right(\frac{\partial P}{\partial x_i}{)}^2{u}_c(x_i{)}^2]}$$(4) (Cui et al. 2018). As the length of the ventilator core (L) and the length of the flow channel cross-section (L_c) were equal, the pressure drop increased as the length of the ventilator core increased. (6) $$\Delta P=C_f\frac{L_c}{D_h}\frac{\rho }{2}{u_r}^2$$(6)

With the help of the channel size of each test core, the hydraulic diameter and Reynolds number of the triangular channel can be calculated using (Gao et al. 2021) (7) $$D_h=\frac{{2W}_c\times H_C}{W_c+2\sqrt{{H_c}^2+{{0.25W}_c}^2}}$$(7) (8) $$Re=\frac{u_c{D}_h{\rho }_{\mathit{air}}}{{\mu }_{\mathit{air}}}$$(8) Figure 8 shows a plot of the friction factor results by core size for H_c = 1.8, 2.0, and 2.6 mm. As the channel velocity increases, the friction factor decreases.

Fig. 8. Friction factor by Reynolds number change within the ventilator core for each ventilator core with the same H_c.

Analysis of the prediction results

Equation 7(7) $$D_h=\frac{{2W}_c\times H_C}{W_c+2\sqrt{{H_c}^2+{{0.25W}_c}^2}}$$(7) shows the equivalent pressure drop (P_e), which is obtained by dividing the pressure drop of the ventilator core (P_c), which is the pressure drop value of the ventilator core for each experimental condition, by the element length L (the length of ventilator core) under the following experimental conditions: (9) $$P_e=\frac{P_c}{L}$$(9)

Figure 9 shows a plot of the value of P_e against the channel velocity (u_c) in the ventilator core. The pressure drops per unit length of ventilator cores with the same H_c and W_c values were similar.

Fig. 9. Equivalent pressure drop changes along the channel velocity.

In the case of ventilator cores with the same cell height (H_c), the P_e value changed linearly with u_c, although when L was different.

Based on the results shown in Figure 9, the ratio of the frontal area (R_fa) was substituted with parameter x₁ and u_c with parameter x₂ to perform multivariate polynomial regression. Parameter x₁ is expressed as a function of R_fa of the ventilator core, as defined in Equation 10(10) $$x_1=\frac{(R_{fa}-R_{fa,L})}{(R_{fa,U}-R_{fa,L})}$$(10) . Parameter x₂ is expressed as a function of the channel velocity in the ventilator core, as defined in Equation 11(11) $$x_2=\frac{(u_c-u_{c,L})}{(u_{c,U}-u_{c,L})}$$(11) . Table 4 lists the values used to obtain the x₁ and x₂ values corresponding to the upper and lower limits of R_fa and u_c.

Table 4. The upper and lower limits of R_fa and u_c.

	Lower	Upper
Rfa	45.9967	47.2285
uc	0.7296	8.9923

Based on the results shown in Figure 9, the multivariate polynomial regression analysis can be expressed in the form of Equation 12(12) $$P_e\left(Pa/m\right)={\propto }_1{x}_1+{\propto }_2{x}_2+{\propto }_3{x}_1^2+{\propto }_4{x}_2^2+{\propto }_5{x}_2^3+{\propto }_6{x}_1{x}_2+{\propto }_7$$(12) , with the coefficient values of α₁–α₇ presented in Table 5. With the R_fa obtained from the shape and thickness of the membrane and spacer according to Equation 2(2) $$R_{fa}=0.5\times \left[1-\left(\frac{W_c{\times \mathrm{T}}_m+2\times \sqrt{H_c\times H_c+0.5W_c\times 0.5W_c}\times T_s}{H_c\times W_c}\right)\right]$$(2) and u_c obtained at an arbitrary airflow rate according to Equation 3(3) $$u_c=u\frac{1}{R_{fa}}$$(3) , the differential pressure was predicted by multiplying the result of Equation 12(12) $$P_e\left(Pa/m\right)={\propto }_1{x}_1+{\propto }_2{x}_2+{\propto }_3{x}_1^2+{\propto }_4{x}_2^2+{\propto }_5{x}_2^3+{\propto }_6{x}_1{x}_2+{\propto }_7$$(12) by the core size. This means that only when the core size (L) and airflow values were determined could the pressure drop in the overall area of a ventilator core be predicted. Multivariate polynomial regression analysis was conducted using the PIAnO 2021 Metamodeler PIDOTECH. (10) $$x_1=\frac{(R_{fa}-R_{fa,L})}{(R_{fa,U}-R_{fa,L})}$$(10) (11) $$x_2=\frac{(u_c-u_{c,L})}{(u_{c,U}-u_{c,L})}$$(11) (12) $$P_e\left(Pa/m\right)={\propto }_1{x}_1+{\propto }_2{x}_2+{\propto }_3{x}_1^2+{\propto }_4{x}_2^2+{\propto }_5{x}_2^3+{\propto }_6{x}_1{x}_2+{\propto }_7$$(12)

Table 5. The results obtained for ${\propto }_i$ s (i = 1–7).

Coefficients	${\propto }_1$	${\propto }_2$	${\propto }_3$	${\propto }_4$	${\propto }_5$	${\propto }_6$	${\propto }_7$
Trained results	86.83	1,360.94	−147.17	1,249.79	−52.57	−663.89	112.53

Equations 13–16 allow the prediction equation and coefficient of determination of the experimental value to be calculated through multivariate polynomial regression analysis. The coefficient of determination (R²) of the theoretical functional formula used in this study was 0.99895. (13) $$R^2=1-\frac{{SS}_{\mathit{err}}}{{SS}_{\mathit{tot}}}=\frac{{SS}_{\mathit{reg}}}{{SS}_{\mathit{tot}}}$$(13) (14) $${SS}_{\mathit{tot}}=\sum _{i=1}^{\mathit{nexp}}(y_i-\overline{y}{)}^2$$(14) (15) $${SS}_{\mathit{reg}}=\sum _{i=1}^{\mathit{nexp}}({\widehat{y}}_i-\overline{y}{)}^2$$(15) (16) $${SS}_{\mathit{err}}=\sum _{i=1}^{\mathit{nexp}}(y_i-{\widehat{y}}_i{)}^2$$(16)

Figure 10 shows a comparison of the functional formula derived through multivariate polynomial regression using the experimental values, most of which were distributed within 5% of the prediction equation.

Fig. 10. A comparison of equivalent pressure drop values through experimental and predicted values.

Conclusions

In this study, experiments were conducted to measure the pressure drop as a function of airflow rate changes by manufacturing nine types of ventilator cores, using ventilator core sizes (0.15, 0.25, and 0.35 mm) and cell heights (1.8, 2.0, and 2.6 mm) as variables. To analyze the pressure drop characteristics of the ventilator core, the R_fa (ratio of frontal area) of the ventilator core to the u_c (channel velocity) inside the ventilator core was calculated. This confirms that the R_fa of an element and the substantial flow rate inside an element are strongly correlated with the characteristics of the differential pressure. The concept of P_e(equivalent pressure drop) was introduced to correct the differential pressure characteristics according to the ventilator core size. In the ventilator cores with the same R_fa, u_r and P_e inside the ventilator core showed a linear relationship, regardless of the ventilator core size. The R_fa of the ventilator core varied depending on the H_c (height of the flow channel cross section).

To analyze the correlation between the R_fa of the ventilator core, channel velocity inside the ventilator core, and equivalent P_e, a multivariate polynomial regression analysis was conducted for three variables to calculate the interaction formula for the equivalent pressure drop. The R² value of the theoretical functional formula is 0.99895. The difference between the derived prediction formula and experimental values was found to be within 5% over the range in which the experiment was conducted.

The interaction formula of P_e, u_c, and R_fa derived in this study can be used to predict the pressure drop across the driving area under conditions including an arbitrary cell height and width of the ventilator core and operating airflow rate.

Nomenclature
u	=	Flow velocity (m/s)
uc	=	Channel velocity (m/s)
V	=	Air flowrate (m3/h)
Rfa	=	Ratio of frontal area
Re	=	Reynolds number
Pc	=	Pressure drop of ventilator core (Pa)
Pe	=	Equivalent pressure drop (Pa/m)
Hc	=	Height of flow channel cross-section (mm)
H	=	Height of ventilator core (mm)
Lc	=	Length of flow channel cross-section (mm)
L	=	Length of ventilator core (mm)
Wc	=	Width of flow channel cross-section (mm)
W	=	Width of ventilator core (mm)
Tm	=	Thickness of membrane (mm)
Ts	=	Thickness of spacer (mm)
yi	=	The ith measured value
$\widehat{y}i$	=	The ith predicted value
R2	=	Coefficient of determination
SStot	=	The total sum of squares
SSreg	=	The regression sum of squares
SSerr	=	The sum of squared errors (residual sum of squares)
ΔP	=	Pressure drop (Pa)
Cf	=	Darcy friction factor
Dh	=	Hydraulic diameter (m)
ρ	=	Air density (kg/m3)
μ	=	Dynamic Viscosity (kg/m·s)

Subscripts
U	=	Upper
L	=	Lower

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education (2021R1I1A3048346).