Design of Draw Resistance of Pressure Drop Standards for Tobacco Products Based on the Flow Distribution

The structural parameters, including capillary diameter and overall length of pressure drop standards were measured using an OGP image measurement device. The pressure drop standard was fixed using a clamping device so that the cross-section of the standard was parallel to the bottom surface of the test bench. OGP's camera focused on the end face of the standard. The experimental instrumentation is shown in Figure 1.

Figure 1.

Optical gaging product (OGP) test instrumentarium.

Commencing from a reference point, the diameters of the ten capillaries were measured in clockwise manner and denoted as d1 to d10 successively. The left panel of Figure 2 illustrates a schematic representation of the end face of the pressure drop standard, while the right panel shows a length diagram of the pressure drop standard. The pressure drop standard was placed horizontally on the test stand and the test software was opened. After that, one end of the standard was searched to draw a boundary line. The camera movement was used to find the other end of the rod to perform the same operation. The software was used to determine the distance between the two lines, i.e. the length of the standard. During empirical testing, we observed that the data from both ends of the pressure drop standard exhibited minimal variance. Consequently, we selected the geometric data from one end for documentation in Table 1. Standard constant flow orifice (CFO) calibration of the pressure drop standard is illustrated in Figure 3. The gas flow rate at the output end of the pressure drop standard controlled by the standard constant flow orifice should be 17.50 ± 0.30 mL/s. Initially, the gas supply switch was turned on, all flow resistances were removed to zero the pressure difference gauge, and the digital differential pressure gauge was zeroed. Then, the pressure drop standard was placed in a dedicated fixture, the air valve was opened, and the pressure difference between the two ends of the rod was tested using a digital differential pressure gauge. Recordings were taken at 2-min intervals, capturing data five times until a state of stability was reached, indicated by no further changes in the reading after 10 min. This final reading represented the draw resistance of the pressure drop standard.

Figure 2.

Geometric structure and schematic diagram of pressure drop standards.

Figure 3.

Schematic diagram of pressure drop standard calibration device based on the critical flow orifice (CFO) method.

The measurement process was repeated three times, and the average reading obtained at the 10-min mark served as the calibrated value for the pressure drop standard. The calibration results were reported with a precision of 0.1 Pa.

The flow of the gas passing through a capillary of the pressure drop standard can be regarded as laminar flow in a circular capillary. After the gas flows into the capillary at a uniform velocity u_m, a boundary layer will be generated near the wall under the action of viscosity. The mass flow through the capillary is constant, and the thickness of the boundary layer increases gradually along the capillary. The velocity of the part that is not affected by the boundary layer will increase until the boundary layer intersects at the center of the capillary cross section, which is the length of the developing section. The flow of the gas after the developing section develops fully. Thus, the total length of the pressure drop standard can be divided into two parts: the developing section L_D and the fully developed section L_developed. The flow distribution in a capillary of the pressure drop standard is shown in Figure 4, where P₀ is the inlet pressure and δ(x) is the thickness of the boundary layer. We have conducted a detailed simulation analysis of the inlet and outlet sections of the capillary, taking into account the disturbances and turbulences that may be caused by sudden contraction or expansion during testing. The simulation results are shown in Figure 5, where Figure 5 (a) displays the velocity cloud diagram of the inlet section, and Figure 5 (b) displays that of the outlet section. It is clear from the figure that there is a fully developed cross-section at the air inlet. In contrast, the turbulence effect at the outlet is not significant. Therefore, in subsequent discussions, we did not consider the turbulence effect at the outlet.

Figure 4.

Schematic diagram of the developing and fully developed flow sections in a capillary of the pressure drop standard.

Figure 5.

Velocity cloud diagrams of the inlet and outlet sections.

The pressure drop along a rough capillary in steady flow of an incompressible viscous fluid is related to the length of the capillary L, the capillary diameter d, the absolute roughness ɛ, the average velocity u_m, and the fluid density ρ. Based on the fundamental principles of fluid mechanics, the pressure drop can be expressed as: [1] ΔP=f(Re, εd)Ldρum22 \Delta P = f({\mathop{\rm Re}\nolimits},\,{\varepsilon \over d}){L \over d}{{\rho u_m^2} \over 2}

Let λ=f (Re, ɛ/d), which is called the energy loss coefficient, the value of which is determined by experiment, then the equation [1] can be expressed as: [2] ΔP=λLdρum22 \Delta P = \lambda {L \over d}{{\rho u_m^2} \over 2}

Equation [2] is applicable to both laminar and turbulent flows. The volumetric flow rate Q is: [3] Q=um⋅A Q = {u_m} \cdot A where A is the cross-sectional area of the capillary, then equation [2] can be expressed as: [4] ΔP=λLdρ2A2Q2 \Delta P = \lambda {L \over d}{\rho \over {2{A^2}}}{Q^2}

The total flow resistance R is defined as: [5] R=λLdρ2A2 R = \lambda {L \over d}{\rho \over {2{A^2}}}

So: [6] ΔP=R⋅Q2 \Delta P = R \cdot {Q^2}

The total flow resistance R is divided into the developing section and fully developed section and can be expressed as: [7] R=RD+Rdeveloped R = {R_D} + {R_{developed}} where R_D represents the flow resistance of the developing section, and R_developed represents the flow resistance of the fully developed section.

According to Wang (8), on the inlet section of circular capillary for laminar and turbulent flows, the length of the developing section can be obtained as: [8] LD=0.0288 Re⋅d {L_D} = 0.0288\,{\mathop{\rm Re}\nolimits} \cdot d where Re is the Reynolds number of the flow, d is the diameter of the capillary.

The pressure loss coefficient λ_p in the developing section can be expressed as: [9] λp=P0−P12ρum2=ΔP12ρum2 {\lambda _p} = {{{P_0} - P} \over {{1 \over 2}\rho u_m^2}} = {{\Delta P} \over {{1 \over 2}\rho u_m^2}} where P₀ is the inlet pressure, P is the pressure at a certain position, ρ is the gas density, and u_m is the gas velocity. Comparing with equation [2], the pressure drop in the developing section can be expressed as: [10] ΔP1=λp⋅12ρum2=λLd⋅12ρum2 \Delta {P_1} = {\lambda _p} \cdot {1 \over 2}\rho u_m^2 = \lambda {L \over d} \cdot {1 \over 2}\rho u_m^2

As shown in Figure 6, the pressure drop from the inlet section to the i-th section in the developing section, which is divided into n segments, can be expressed as: [11] ΔPi=λpi⋅12ρum2=∑j=1nλjLj−Lj−1d⋅12ρum2=∑j=1nλjΔLjd⋅12ρum2 \matrix{{\Delta {P_i} = {\lambda _{pi}} \cdot {1 \over 2}\rho u_m^2 = \sum\limits_{j = 1}^n {{\lambda _j}{{{L_j} - {L_{j - 1}}} \over d} \cdot {1 \over 2}\rho u_m^2}} \hfill \cr {= \sum\limits_{j = 1}^n {{\lambda _j}{{\Delta {L_j}} \over d} \cdot {1 \over 2}\rho u_m^2}} \hfill \cr}

Figure 6.

Schematic diagram of segmentation in the developing section.

The relationship between the pressure loss coefficient λ_pi from the inlet to the i-th segment and the pressure loss coefficient λ_i of the i-th segment can be expressed as: [12] λpi=∑j=1nλjΔLjd {\lambda _{pi}} = \sum\limits_{j = 1}^n {{\lambda _j}{{\Delta {L_j}} \over d}} where λ_p can be obtained by looking up the corresponding table in (8). When the thickness of the boundary layer reaches its maximum, i.e., the junction between the developing section and the fully developed section, we have: [13] λp=∑i=1nλiΔLid=4 {\lambda _p} = \sum\limits_{i = 1}^n {{\lambda _i}{{\Delta {L_i}} \over d} = 4}

Combined with equation [7], the flow resistance of the developing section can be expressed as: [14] RD=(∑i=1nλiΔLid)ρ2A2=4ρ2A2 {R_D} = \left( {\sum\limits_{i = 1}^n {{\lambda _i}{{\Delta {L_i}} \over d}}} \right){\rho \over {2{A^2}}} = 4{\rho \over {2{A^2}}}

The flow in the developed section is laminar. According to Nikuradse (9), in laminar flow, the friction coefficient varies only with the Reynolds number, regardless of relative roughness, and the method for calculating the friction coefficient along the length of the capillary is: [15] λdeveloped=64Re {\lambda _{developed}} = {{64} \over {{\mathop{\rm Re}\nolimits}}}

Combined with the calculated length L_D in equation [8], the flow resistance of the fully developed section can be expressed as: (16) Rdeveloped=λdevelopedd(L−LD)ρ2A2 {R_{developed}} = {{{\lambda _{developed}}} \over d}(L - {L_D}){\rho \over {2{A^2}}} where L represents the total length of the pressure drop standard.

Based on the above analysis, an iterative calculation algorithm was established, as shown in Figure 7. In accordance with the previously mentioned resistance theory, given the pressure drop value, we could verify the value by calculating with the theoretical capillary diameter. To validate the accuracy of the theory, seven pressure drop standards were created. An iterative approach was used to guide the allowable deviation in the theoretical diameter of the pressure drop standards and to instruct the deviation of the pressure drop standards based on the range of diameter variations. Given that the total flow rate was constant and as an initial condition for the iteration, the flow rate was set so that it was equally distributed across the capillaries. However, due to slight differences in the diameter and length of each capillary, after one iteration, the flow rate for each capillary would also differ. Therefore, multiple iterations were carried out to obtain more accurate flow rates for each capillary and ultimately to calculate a more precise value of draw resistance. This algorithm was capable of converging to a precise value, primarily due to its explicit optimization objective: the pressure drops of the pressure drop standards. In each iteration, the algorithm updates the flow rate of each capillary to calculate the pressure drop of the pressure drop standard. Moreover, the algorithm's termination condition was set such that when the difference between the pressure drops in two consecutive iterations was less than 0.001, the algorithm ceased to iterate. This termination condition is a typical criterion for convergence. If this condition is met within a finite number of iterations, we could assert that the algorithm had indeed converged.

Figure 7.

Flow chart of the computational algorithm. Where N represents the number of capillaries and A_k represents the cross-sectional area of each capillary.

The program was designed according to the MVVM (Model-View-View model) pattern. The view was responsible for the appearance of the interface of the entire program, the view model was responsible for modeling the content presented in the interface, and the model was also used to store data. The interface was developed using WPF (Windows Presentation Foundation), the latest generation of Microsoft's graphics system, and the logical functions of the program were implemented in C#.

The program allows for the input of experimental environmental parameters (temperature, humidity, atmospheric pressure) and reads the diameter and length of the ten capillaries of the pressure drop standard obtained from OGP testing in a comma-separated values (CSV) format. Following the logic outlined in theoretical analysis, the program performed iterative calculations.

The iteration process was guided by the criterion that the difference in pressure drop between two consecutive steps should be below 10⁻³ Pa or reached the maximum number of iterations, upon which the calculation loop was terminated. Ultimately, the program computed the draw resistance of the pressure drop standard, the flow rate for each capillary, and the proportion of the development length to the total length. The results were then displayed in the interface, providing a comprehensive presentation of the calculations performed.