3.1 Transient vs. steady airflow
The transient inspiratory and expiratory streamlines patterns (see , t = 1/10T and , 9/10T, restful breathing condition, where T is the cycling period) are similar to those from quasi-steady flow simulation (Kimbell et al., 1997
; Yang, Scherer, & Mozell, 2007
) and flow visualization (Morgan et al., 1990). In briefly, flow streams released from dorsal medial side of the external naris traced out S-shape curves and passed through the olfactory recess during inspiration. Those released from ventral medial side flowed along the floor of the nasal cavity. Streamlines passing through the dorsal lateral meatus and ventral lateral meatus originated from the dorsal lateral side and ventral lateral side of the external naris. During expiration, much less streamlines entered the olfactory region than inspiratory flow.
Figure 2 Streamlines in rat nasal cavity during (a) acceleration, inspiration (1/10T) and (b) deceleration, expiration (9/10T); and axial velocity distributions at (c)1/10T and (d) 9/10T in 4 coronal sections (section1- section 4) as indicated in Figure 2(a–b) (more ...)
For both inspiration and expiration (see , t = 1/10T and , 9/10T, restful breathing condition), the primary axial airflow occurs ventrally, in the nasopharyngeal meatus, with a second peak velocity in the dorsal recess, which are consistent with previous steady state studies (Kimbell et al., 1997
; Yang, Scherer, & Mozell, 2007
). The axial velocity distribution patterns were quantitatively and qualitatively compared at four coronal planes and found to be similar between transient inspiratory acceleration (t=1/10 T), deceleration (t=4/10T) and steady state, all of which have the same flow rate. The differences of velocity at the four planes between the three phases were evaluated by the normalized mean square error (NMSE) (ASTM, 2002
). The NMSE values were calculated to be less than 0.07, indicating good agreement between the velocity profiles of the three phases. Same observations apply for the expiratory phase at t=6/10T and 9/10T, and for the two other sniffing conditions. Thus, it seems that the transient effects on rat nasal airflows are negligible and the flow field can be approximated as quasi-steady. However, significant differences of velocity distribution were found between acceleration phase, deceleration phase, and steady case with equivalent flow rate near the transition between respiratory phases, for example, t=1/100T or 99/100T, indicating the non-negligible effects of flow unsteadiness.
What has not been described in details in previous literature are the secondary motion of rat nasal flows, which are induced by curvature, directional changes and rapid expansion of the nasal passages. The secondary flow would extract momentum from the primary (axial) flows, reduce primary velocity, increase the residence time of the inhaled nanoparticles, enhance the lateral mixing of the mass and momentum, and therefore, strongly affect the nanoparticle deposition in the nasal cavities. shows the secondary flow (indicated by arrows) for the inspiratory acceleration (, t=1/10T) and expiration deceleration (, t=9/10T) during restful breathing. In the anterior part the nose (plane1), secondary flows were predicted to move away from septum and flow into meatus during inspiration but move towards to septum and/or flow out of meatus during expiration. In the posterior regions, where the olfactory recess is mostly located (plane 2–4), the secondary flows were found to move mainly downwards towards the nasal floor during inspiration and upwards towards the nasal roof during expiration. The strength of the secondary flow (SS
) is defined as the ratio of secondary flow velocity to total velocity,
, where u
is the velocity component in the main (axial) flow direction, and v
are the velocity components in the plane perpendicular to u
. shows that SS
is consistently stronger in the posterior regions than the anterior ones during both inspiration and expiration, which may affect the nanoparticle deposition in the olfactory region. In accordance with to axial velocity distribution, secondary flow fields during acceleration (t=1/10T) and deceleration (t=4/10T) are quite similar to those of steady state flow with equivalent flow rate in the region far away from transition between respiratory phases.
Vector plots and strength iso-contours of secondary flow in coronal sections (section1- section 4) at (a) 1/10T (inspiration; top) (b) 9/10T (expiration; bottom) during restful breathing.
The effects of unsteadiness on the developing flows in finite length channels can be estimated by the Womersley number W0
) and the Strouhal number S
(Schroter & Sudlow, 1969
where R is the hydraulic diameter of the external naris, f
is the cycling frequency, ν is the air kinematic viscosity, L is the axial length along the nasal cavity, and U is the average velocity during one cycle or instantaneous velocity at the external naris. By definition, the Womersley number is a dimensionless ratio of the unsteady inertial forces in relation to viscous forces. Whereas, the Strouhal number, which is derived from the ratio of the steady boundary layer thickness to the Stokes layer thickness (Pedley, Schroter, & Sudlow, 1977
), represents the ratio of unsteady inertial forces or local acceleration to the convective inertial forces. Quasi-steady flow can be assumed as long as W0
is less than 4 and S
is less than 1 (Isabey & Chang, 1981
). For the present rat nasal airway, the values of W0
based on average velocity were estimated to be 1.45, 2.64, 3.73 and 0.33, 0.53, 0.54 for restful breathing, moderate sniffing and strong sniffing, respectively. Although these values seems to meet the criteria of flow quasi-steadiness (W0
<4 and S
<1), the aforementioned flow unsteadiness near the respiratory transition indicates that Strouhal number based on average velocity can not adequately characterize flow states. Instead, Strouhal number based on instantaneous velocity is more appropriate, as shown in , which is a Moody plot of non-dimensionalized pressure drop versus the Reynolds number for the three unsteady respiratory conditions and steady flows during inspiration () and expiration (. It is evident from that unsteadiness is significant only near the transition between inspiration and expiration, and negligible near peak flow. More specifically, the quasi-steady assumption is valid for about 84%, 76%, and 72% of the cycling period for restful breathing, moderate sniffing, and strong sniffing respectively. Furthermore, also shows that resistance during the deceleration is lower than during the acceleration for both inspiratory and expiratory flows when the unsteady effects are significant, indicating energy dissipation during the respiratory cycle. Such hysteresis of pressure loss has been reported in the human rhinomanometry measurements (Sullivan & Chang, 1991
). Finally indicates that, near the transition of respiratory phases (smaller Re number), unsteady flows during the deceleration deviate more from steady flows than those during acceleration for both inspiration and expiration.
Figure 4 Moody diagram of the non-dimensionalized pressure drop vs. Reynolds number. The data are plotted for both inspiratory flow (a, top plot) and expiratory flow (b, bottom plot) under the three unsteady respiratory breathing conditions (Restful breathing: (more ...)
3.2 Transient vs. steady nanoparticle transport and deposition
We applied the same strategy to compare the nanoparticle transport and deposition at various acceleration phases (for example, t = 1/10T and 2/10T), deceleration phases (for example, t =3/10T and 4/10T) and steady state to examine the unsteady effects. If the quasi-steady assumption for particle transport would hold, there wouldn’t be any significant difference in either the deposition flux or concentration field between these paired conditions, which all has the same flow rate. However, the differences observed (, deposition flux of 1 nm particles on the septal wall) are significant for t=1/10T vs. t=4/10T (), with more particles reaching the downstream during deceleration (t=4/10T) than that during acceleration (t=1/10T) and the steady state. The differences are likely to be resulted from temporal separation of momentum and mass transport that more nanoparticles are inhaled into nasal cavity during the interval between t=1/10T and t=4/10T due to the higher flow rate than that during the interval of 0~1/10T, which results in accumulation of particles within the airstream, and subsequently higher deposition. The accumulation effect weakens near the peak of inspiration, for example, at t=2/10T and. t=3/10T, where the particle deposition pattern and concentration is quite similar to that of quasi steady state (). Similar effects were also observed during the expiratory phase. Thus, the quasi-steady assumption of 1 nm nanoparticle transport in rat nasal cavity at restful breathing is only valid in the temporal period far away enough from the respiratory transition, in other words, near the vicinity of respiration peaks. This is similar to flow field but the quasi-steadiness temporal period of nanoparticles is shorter, as seen from above observations. Furthermore, the quasi-steadiness period becomes even shorter with increase of nanoparticle size and changing from restful breathing to moderate and strong sniffing. For example, when particle size increases from 1 nm to 2.78 nm for the restful breathing, or when restful breathing changes to moderate sniffing for 1 nm particles, the differences of concentration distribution between t=2/10T (or 3/10T) and steady case become significant. Within a even narrower temporal window, for example, t=2.3~ 2.7/10T for moderate sniffing for 1 nm particles, and t=2.45 ~ 2.65/10T for strong sniffing for 2.78 nm particles, good agreement can still be found for the unsteady and equivalent steady state in concentration field.
Figure 5 Deposition flux of 1 nm particles on the nasal cavity during inspiration at airflow rate of 74 mL/min (Left column; during acceleration--t=1/10T, deceleration--t=4/10T, and steady state) and 119 mL/min (Right column; during acceleration--t=2/10T, deceleration--t=3/10T, (more ...)
The validity of quasi-steadiness of nanoparticle field is also spatially dependent. For example, the concentration of 100 nm particles at the peak inspiration during strong sniffing was significantly different from that of steady case with equivalent flow rate in the posterior nasal region (e.g. plane 4, see or ). On the other hand, the concentration distributions of unsteady and steady cases in the anterior nasal region (plane 1, see or ) were still similar, indicating the validity of quasi-steadiness of particle field in this region. Furthermore, it was found that this spatial region expands from anterior towards posterior nasal cavity with decrease of particle size. Similar trend was found when changing from moderate sniffing to restful breathing or from strong sniffing to moderate sniffing.
3.3 Transient vs. steady nasal deposition dosimetry as a function of nanoparticle size and breathing conditions
CFD simulations were used to calculate the deposition dosimetry as a function of particles size and breathing rate, a further comparison of the unsteady effects. summarizes such function for the whole nasal cavity () and olfactory region () for both steady and unsteady state. The unsteady state deposition was averaged over the inhalation cycle and the average flow rate for the given inhalation cycle was used for the steady case. The deposition efficiency for whole nasal cavity decreases with the increase of particle size and flow rates for both cyclic and steady cases. Particle diffusivity is reversely proportional to the particle size so larger nanoparticles have lower deposition efficiency. Increase of breathing rates decreases the residence time of nanoparticles in the nasal cavity thus reduces the opportunities to deposit onto the wall. Thus larger deposition efficiency is found for lower breathing flow rate. While, the deposition efficiency for cyclic flow is lower than those for steady case. The reduction increases with nanoparticle size and flow rate. For example, deposition efficiency in cyclic restful breathing condition of 1 nm, 10 nm, and 100 nm nanoparticles decreases about 4%, 20%, and 40%, as compared to their equivalent steady state flow.
Comparison the total deposition efficiency between transient and equivalent steady case in the whole nasal cavity (a, left) and Olfactory region (b, right).
For deposition to the olfactory region, there exists an optimal particle size for maximum deposition rate. The optimal size decreases with the increase of flow rate. For example, for steady conditions, it decreases from 5 nm at restful breathing to 3 nm at strong sniffing. The increase of flow rate results in higher deposition efficiency before maximal deposition but the trend is reversed after maximal deposition. Similarly, deposition efficiency for the cyclic case is slight higher (lower) than those of steady case before (after) reaching the maximal values. The above observation can be explained by the counter-acting effects of particle diffusivity, particle concentration remaining in the airstream when entering the olfactory region, and residence time of particles in the olfactory region. For a given flow rate, smaller nanoparticles (for example, 1 nm or 2 nm) deposit more in the anterior region than large nanoparticles (3 nm, 5 nm or 100 nm) and thus less remaining in the airstream when enter the olfactory region. On the other hand, larger nanoparticles are less diffusive than smaller nanoparticles and hence have less deposition efficiency than smaller nanoparticles. The counter-balancing effects resulted in an optimal particle size for maximal deposition. Similarly, as the flow rate increases, less nanoparticles deposit in the anterior region and more nanoparticles remain in the airflows entering the olfactory region; while the particle residence time in the olfactory region decreases with increase of flow rate. These effects combined resulted in the increase of deposition efficiency in the olfactory region for smaller nanoparticles but decreases for large ones with the increase of flow rate. Thus, smaller optimal particle size is reached for higher flow rates.
One reason for the reduction of particle deposition efficiency during cyclic flow is the increase of flow rate near the peak flow. Within the inspiratory cycle, the flow rate in the period between t=0.22T and t=0.39T is higher than that steady flow rate which is about 64% of peak flow rate of cyclic flow. Using the deposition equations for steady flow (See Equation (5)
in the following text ) and assuming that particle deposition in infinitely small time is steady, integration of instantaneous deposition over the inspiratory period of a cycle during moderate sniffing gives deposition efficiency lower than steady case but higher than cyclic case (). This indicates that the increase of flow rate near the peak inspiration can not adequately account for the reduction of particle deposition during cyclic flows. It should be noted that the above integration assumes the instantaneous particle deposition is in the same phase with cyclic flows.
depicts the variation of instantaneous deposition (normalized by peak value) with particle size and flow rate during the inspiration phase of one cycle. During restful breathing for 1 nm nanoparticle, the position of maximum instantaneous deposition rate is very close to that of peak inspiration (t=T/4). With increase of particle size and/or change of respiratory condition from restful breathing to sniffing, the position of maximal deposition rate moves away from peak inspiration and towards deceleration phase. At the same time, the instantaneous deposition rate curve becomes highly unsymmetrical and much higher deposition rate is found during the deceleration than acceleration phase. At the end of inspiration, the instantaneous deposition rate doesn’t decrease to zero and takes higher value for larger particles and/or stronger respiratory conditions. The deposition rate during the acceleration is lower than the deceleration especially for larger nanoparticles and higher flow rate/sniffing frequency. also explains why decreased diffusion effects near peak inspiratory flow alone cannot account for the deduction of deposition during cyclic breathing conditions: the particle deposition curve deviates from the inflow pattern (sinusoid) with increase of particle size and/or changing from restful breathing to sniffing conditions. In other words, the phase shift (defined as time interval between the point when peak flow rate is reached and the one when deposition rate reaches the maximal) and non-zero deposition at the start/end of inspiration increase with particle size and breathing conditions.
Normalized depositions during inspiration phase in one cycle.
Changing from restful breathing to sniffing is associated with an increase in both flow rate and breathing frequency (). It is desirable to investigate the individual effects of these two factors on the particle deposition and transport. A series of computations (results not shown here), assuming sinusoidal inflow, were conducted for 10 nm particles at constant frequency (5 Hz) but variable peak flow rates (100~1000 ml/min), and constant peak flow rate (252 ml/min) but variable frequency (1.5~10 Hz). It was found that phase shift increased with frequency but decreases with peak flow rate. Combined with increasing effects of particle size under the same breathing condition (restful breathing, moderate sniffing, or strong sniffing) shown in , the phase shift is the net result of competing effects of particle size, peak flow rate, and cycling frequency, or in non-dimensional form, proportionally related to the particle Strhouhal number (StrC Sc1/3S
, where Sc
is the Schmidt number (ν/
D) and D is the diffusion coefficient of particle in air, Grotberg, Sheth, & Mockros, 1990
). Nonetheless, the phase shift is not determined by the StrC
number alone but also the depends on the Str
number (Equation (5)
). shows that strong sniffing has larger phase shift and non-zero deposition at the start/end of inspiration than moderate sniffing although they have similar StrC
numbers for given particle sizes (). Noting that both flow rate and frequency were doubled from moderate sniffing to strong sniffing and the value of W0
number during strong sniffing was larger than during moderate sniffing, thus the variations in frequency more significantly affected the phase shift than flow rate.
Average Particle Strouhal number ( StrC )
The potential value and future application for this study lies in if we can use CFD models to calculate the deposition dosimetry as a function of particles size and breathing conditions, with correction for the unsteady effects. Thus, empirical equations are then fitted to characterize the nanoparticle deposition rates in both whole nasal cavity and olfactory region taking into account the steady and cyclic breathing conditions. The equation for whole nasal cavity deposition function takes the form of:
Where D is the particle diffusivity, Q is the flow rate, Ci
(i=1, 2, 3) is fitted coefficient. By fitting equation (6)
to the simulated deposition data, values of Ci
(i=1, 2, 3) were obtained with mean and standard deviation as follows: C1
=12.26±0.2, C3=1.47±0.06 for steady flow, and C1
=1.47±0.07 for cyclic flow. The R2
values for the fit are greater than 0.99 (). In a similar way, Cheng et al. (1990)
and Garcia and Kimbell (2009)
derived the deposition models for nanoparticles through the rat nasal cavity during only steady inspiratory flows based on either experimental data and simulated results, where slightly different equation forms (compared with (6
)) were adopted: D0.517Q−0.234
for Cheng et al (1990)
in Garcia and Kimbell (2009)
. The difference in the diffusion parameter may be attributed to the different nasal cavities adopted in the studies. For example, pharynx and larynx were included in the rat nasal cavities of Cheng et al (1990)
but not in our model and that of Garcia and Kimbell (2009)
Figure 8 Comparisons of nanoparticle deposition efficiency in the whole nasal cavity (a, left) and olfactory region (b, right) under cyclic breathing conditions and those with fitted equations (6) and (7) respectively.
The regional deposition in the olfactory region can be fitted with the log-normal equation:
The values of Ci (i=1,2, 3) are calculated as C1=3.878±0.512, C2=−0.051±0.004, C3=16.585±1.517 for steady flow, and C1=4.636±0.824, C2=−0.048±0.005, C3=10.874±1.229 for cyclic flow. The R2 values for the fit are greater than 0.96 ().
For future applications, these fitted equations (6)
with appropriate set of C values can be potentially used to derive deposition efficient for a wide range of particle size and sniffing conditions under both unsteady and steady flow conditions, without changing the base form of the equation.
Since unsteady simulations of nasal airflow and nanoparticle deposition are much more computationally expensive than steady ones, it would be valuable to derive an empirical effective flow rate that can estimate the averaged nanoparticle deposition efficiency during breathing cycles based on steady case of that effective flow rate. From , it can be concluded that such effective flow rate (matching flow rate) should be higher than the average cyclic flow rate but lower than peak inspiration flow rate:
For micro-particles deposition in a triple bifurcation lung airway model (Zhang & Kleinstreuer, 2004
) and for nanoparticle deposition in a human nose (Shi, Kleinstreuer, & Zhang, 2006
), previous studies have derived C to be a constant of 0.5. Based on our results, the value of C varies with particle size: 0.5 for smaller nanoparticles (<1 nm), but decrease to 0.4 for larger nanoparticles (>1 nm). The agreement between simulation and prediction with matching flow rate is good (see ): the R2
values for the prediction are greater than 0.99 in the whole nasal cavity () and 0.95 in the olfactory region (). A comparison of and shows that prediction with matching flow rate agrees slightly better with simulation than empirical equation (7)
in the olfactory region.
Comparisons of nanoparticle deposition efficiency in the whole nasal cavity (a, left) and olfactory region (b, right) under cyclic breathing and effective steady conditions.
3.5 Turbulent or laminar rat nasal airflows
The above simulations were conducted under the assumption that rat nasal airflows are laminar flows. Although the maximal Reynolds number based on velocity at the external naris and its diameter is less than 2300 for the respiratory conditions considered in this study, this can not guarantee the laminar nature of rat nasal airflow. The complicated anatomy of rat nasal passages consisting of diverging and converging conduits may make the transition from laminar flows to turbulent flows to occur at a Reynolds number considerably below the critical value of 2300. The pressure drop measured by Cheng et al (1990)
in rat nasal casts was found to vary nonlinearly with flow rate. However, this non-linearity in the pressure-flow rate relationship can not exclude the validity of laminar assumption either since laminar flow can also show such nonlinear variation. To evaluate possible turbulent effects in the rat nasal cavity, inspiratory airflow during steady strong sniffing () was simulated with the Large Eddy Simulation (LES) with dynamical subgrid model (Germano, Piomelli, Moin, & Cabot, 1991
) implemented in Fluent (Ansys, Inc., Canonsburg, PA). The advantage of dynamical LES over traditional RANS turbulence models (Wilcox, 1998
) lies in its ability for accurate prediction of the full range of flow types, laminar, transitional, and turbulent flows (Germano, Piomelli, Moin, & Cabot, 1991
). Grid independence study was performed to determine an acceptable grid resolution to use for strong sniffing. The grid with 2,116,059 elements, which resolves over 80% of the turbulent kinetic energy through the nasal cavity, produced similar pressure drop across the nasal cavity and velocity profiles at certain locations as that with 1,782,194 elements, indicating an adequate grid resolution (Pope, 2000
). The results from 2,116,059 element grid simulation showed that velocity distributions derived from dynamical LES were basically identical to those from laminar model. Furthermore, even imposing some turbulent fluctuations in the external naris, they diminished very quickly in the anterior skin region and were basically zero in the main nasal cavity. Since the flow rate of strong sniffing is highest among the three respiratory breathing conditions considered in this study () and transition to turbulence only occurs at high flow rate, the laminar flow assumption is also valid for the moderate sniffing and restful breathing.