Flow and pressure drop in systems of repeatedly branching tubes

1971 ◽  
Vol 46 (2) ◽  
pp. 365-383 ◽  
Author(s):  
T. J. Pedley ◽  
R. C. Schroter ◽  
M. F. Sudlow
Keyword(s):  
Pressure Drop ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Average Value ◽  
Two Generations ◽  
Mean Pressure ◽  
Branching Angle ◽  
The Mean ◽  
Set Up

The airways of the lung form a rapidly diverging system of branched tubes, and any discussion of their mechanics requires an understanding of the effects of the bifurcations on the flow downstream of them. Experiments have been carried out in models containing up to two generations of symmetrical junctions with fixed branching angle and diameter ratio, typical of the human lung. Flow visualization studies and velocity measurements in the daughter tubes of the first junction verified that secondary motions are set up, with peak axial velocities just outside the boundary layer on the inner wall of the junction, and that they decay slowly downstream. Axial velocity profiles were measured downstream of all junctions at a range of Reynolds numbers for which the flow was laminar.In each case these velocity profiles were used to estimate the viscous dissipation in the daughter tubes, so that the mean pressure drop associated with each junction and its daughter tubes could be inferred. The dependence of the dissipation on the dimensional variables is expected to be the same as in the early part of a simple entrance region, because most of the dissipation will occur in the boundary layers. This is supported by the experimental results, and the ratio Z of the dissipation in a tube downstream of a bifurcation to the dissipation which would exist in the same tube if Poiseuille flow were present is given by \[ Z = (C/4\surd{2})(Re\,d/L)^{\frac{1}{2}}, \] where L and d are the length and diameter of the tube, Re is the Reynolds number in it, and the constant C (equal to one for simple entry flow) is equal to 1·85 (the average value from our experiments). In general, C is expected to depend on the branching angles and diameter ratios of the junctions used. No experiments were performed in which the flow was turbulent, but it is argued that turbulence will not greatly affect the above results at Reynolds numbers less than and of the order of 10000. Many more experiments are required to consolidate this approach, but predictions based upon it agree well with the limited number of physiological experiments available.

Steady Flow Through a Double Converging-Diverging Tube Model for Mild Coronary Stenoses

1989 ◽  
Vol 111 (3) ◽  
pp. 212-221 ◽  
Author(s):  
S. C. van Dreumel ◽  
G. D. C. Kuiken
Keyword(s):  
Pressure Drop ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Tube Model ◽  
Mean Flow ◽  
Number Range ◽  
Low Reynolds Numbers ◽  
In Series ◽  
Coronary Stenoses ◽  
Converging Angle

Velocity profiles and the pressure drop across two mild (62 percent) coronary stenoses in series have been investigated numerically and experimentally in a perspex-tube model. The mean flow rate was varied to correspond to a Reynolds number range of 50–400. The pressure drop across two identical (62 percent) stenoses show that for low Reynolds numbers the total effect of two stenoses equals that of two single stenoses. A reduction of 10 percent is found for the higher Reynolds numbers investigated. Numerical and experimental results obtained for the velocity profiles agree very well. The effect of varying the converging angle of a single mild (62 percent) coronary stenosis on the fluid flow has been determined numerically using a finite element method. Pressure-flow relation, especially with respect to relative short stenoses, is discussed.

Pressure drop and power dissipation in oscillatory wavy-walled-tube flows

1988 ◽  
Vol 187 ◽  
pp. 573-588 ◽  
Author(s):  
M. E. Ralph
Keyword(s):  
Pressure Drop ◽  
Flow Rate ◽  
Power Dissipation ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Frequency Parameter ◽  
Mean Power ◽  
Pressure Drops ◽  
The Mean

Pressure drops occurring in oscillatory viscous flows in wavy-walled tubes have been studied experimentally, for Reynolds numbers up to 1500 and Strouhal numbers in the range 0.005 to 0.02, and by numerical solution of the Navier-Stokes equations, for Reynolds numbers up to 200 and Strouhal numbers between 0.005 and 0.1. Agreement was good for values of the mean modulus of the pressure drop at lower Strouhal numbers and for values of the mean power dissipation at all Strouhal numbers.Numerical solutions have shown that the pressure drop may vary non-sinusoidally, even though the imposed variation in flow rate is sinusoidal. This cannot be explained by the nonlinearity of the steady pressure drop-flow rate relationship, and arises because the velocity field is not quasi-steady. In particular energy may be stored in strong vortices formed during the acceleration phase of the flow cycle, and partially returned to the main flow later. The peak pressure drops in such flows, which are associated with the formation of these vortices, can be almost twice as large as values predicted by adding the appropriate quasi-steady and unsteady inertial contributions. This finding is important in the wider context of unsteady conduit flow.The dependences of the mean modulus of the pressure drop and the mean power dissipation on the Strouhal number and frequency parameter were investigated in detail numerically for two geometries. It was not possible to reduce either dependence to a function of a single parameter. The ‘equivalent’ straight-walled tube for power dissipation was found to have a smaller bore than that for pressure drop, leading to smaller ‘phase angles’ than might have been expected at large values of the frequency parameter. This is because as the pressure drop becomes increasingly dominated by unsteady inertia, there remain relatively large recirculations in which energy is dissipated.

Flow Distribution and Pressure Drop in Steady Airflow Through a Selective Laser Sintered Human Tracheobronchial Airway Model

2003 ◽  
Author(s):  
Kah-Hoe Tan ◽  
Ramkumar N. Parthasarathy ◽  
M. Cengiz Altan ◽  
David L. Johnson ◽  
R. E. Clinkenbeard
Keyword(s):  
Pressure Drop ◽  
Static Pressure ◽  
Flow Distribution ◽  
Reynolds Numbers ◽  
Flow Rates ◽  
Physiological Flow ◽  
Static Pressure Difference ◽  
The Mean ◽  
The Relationship ◽  
Airway Model

The flow distribution and pressure drop of steady airflow in the human central airways were studied experimentally using an anatomically correct, selective laser sintered (SLS) human tracheobronchial airway model. Measurements were made for tracheal flow rates ranging from 0.1 to 2.67 liters per second, which correspond to normal physiological flow ranges. The mean air velocities at the exit orifices of the airway model were detected by means of a pitot static tube connected to a pressure transducer. The flow rates, the average velocities, and the Reynolds numbers in each branch of the airway model were then computed. In addition, the static pressure difference between the trachea and the airway exits was measured. The experimental measurements were used to determine the relationship between pressure drop and flow rate. The ratio of inlet to total exit area of the model was identified as a significant factor that influenced the pressure drop. The results obtained in the present study will be particularly useful for validating computational studies.

Steady Inspiratory Flow in a Model Symmetric Bifurcation

1994 ◽  
Vol 116 (4) ◽  
pp. 488-496 ◽  
Author(s):  
Yao Zhao ◽  
Baruch B. Lieber
Keyword(s):  
Test Section ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Axial Position ◽  
Flow Loop ◽  
Low Velocity ◽  
Cross Sectional ◽  
Branching Angle ◽  
Head Pressure ◽  
Constant Head

Flow in a bifurcating tube system typifying a major bronchial bifurcation is studied experimentally with a two color, two velocity component laser Doppler anemometer. The flow loop is composed of a pumping station, flow stratifiers and a constant head pressure tank; it can accommodate steady, pulsatile or oscillatory flow. The test section is a symmetric bifurcation of constant cross sectional area and has a branching angle of 70 deg. The test section is a cast of clear silicon rubber in a plexiglass mold that was milled on a numerically controlled milling machine. The flow division ratio from the parent to daughter branches is about unity. Steady flow results that model the inspiratory phase at Reynolds numbers of 518, 1036 and 2089, corresponding to Dean numbers of 98, 196 and 395, show that in the bifurcation plane velocity profiles in the daughter branches are skewed toward the inner wall. In the transverse plane, “m” shaped velocity profiles are found with low velocity at the center. Secondary flow patterns, which are responsible for such phenomena, are first observed at the axial position where the flow begins to turn. Flow separation was not observed at any point in the bifurcation.

Mean-flow scaling of turbulent pipe flow

1998 ◽  
Vol 373 ◽  
pp. 33-79 ◽  
Author(s):  
MARK V. ZAGAROLA ◽  
ALEXANDER J. SMITS
Keyword(s):  
Friction Factor ◽  
Friction Velocity ◽  
Outer Region ◽  
Reynolds Numbers ◽  
Velocity Profiles ◽  
Mean Velocity ◽  
Velocity Scale ◽  
The Mean ◽  
Log Law ◽  
High Reynolds

Measurements of the mean velocity profile and pressure drop were performed in a fully developed, smooth pipe flow for Reynolds numbers from 31×103 to 35×106. Analysis of the mean velocity profiles indicates two overlap regions: a power law for 60<y+<500 or y+<0.15R+, the outer limit depending on whether the Kármán number R+ is greater or less than 9×103; and a log law for 600<y+<0.07R+. The log law is only evident if the Reynolds number is greater than approximately 400×103 (R+>9×103). Von Kármán's constant was shown to be 0.436 which is consistent with the friction factor data and the mean velocity profiles for 600<y+<0.07R+, and the additive constant was shown to be 6.15 when the log law is expressed in inner scaling variables.A new theory is developed to explain the scaling in both overlap regions. This theory requires a velocity scale for the outer region such that the ratio of the outer velocity scale to the inner velocity scale (the friction velocity) is a function of Reynolds number at low Reynolds numbers, and approaches a constant value at high Reynolds numbers. A reasonable candidate for the outer velocity scale is the velocity deficit in the pipe, UCL−Ū, which is a true outer velocity scale, in contrast to the friction velocity which is a velocity scale associated with the near-wall region which is ‘impressed’ on the outer region. The proposed velocity scale was used to normalize the velocity profiles in the outer region and was found to give significantly better agreement between different Reynolds numbers than the friction velocity.The friction factor data at high Reynolds numbers were found to be significantly larger (>5%) than those predicted by Prandtl's relation. A new friction factor relation is proposed which is within ±1.2% of the data for Reynolds numbers between 10×103 and 35×106, and includes a term to account for the near-wall velocity profile.

Peristaltic Waves in Circular Cylindrical Tubes

1969 ◽  
Vol 36 (3) ◽  
pp. 579-587 ◽  
Author(s):  
F. Yin ◽  
Y. C. Fung
Keyword(s):  
Pressure Gradient ◽  
Wall Motion ◽  
Amplitude Ratio ◽  
Cylindrical Tube ◽  
Critical Value ◽  
Mean Flow ◽  
Navier Stokes Equation ◽  
Mean Pressure ◽  
The Mean ◽  
Set Up

Peristaltic pumping in a circular cylindrical tube is analyzed. The problem is a viscous fluid flow induced by an axisymmetric traveling sinusoidal wave of moderate amplitude imposed on the wall of a flexible tube. A perturbation method of solution is sought. The amplitude ratio (wave amplitude/tube radius) is chosen as a parameter. The nonlinear convective acceleration terms in the Navier-Stokes equation is retained. The governing equations are developed up to the second order in the amplitude ratio. The zeroth-order terms yield the classical Poiseuille flow, the first-order terms yield the Sommerfeld-Orr equation. If there is no pressure gradient in the absence of wall motion, the mean flow and mean pressure gradient (averaged over time) are both shown to be proportional to the square of the amplitude ratio. Numerical results are obtained for this simple case by approximating a complicated group of products of Bessel functions by a polynomial. The results show that the mean axial velocity is dominated by two terms. One term corresponds to a parabolic profile which is due to the mean pressure gradient set up by the wall motion. The other term arises from satisfying the no-slip boundary condition at the wavy wall rather than at the mean position of the wall. In addition, there are perturbations arising from the convective acceleration. If the mean pressure gradient set up by the wall motion itself reaches a certain positive critical value, the velocity becomes zero on the axis. Values of the mean pressure gradient larger than the critical value will induce backward flow in the fluid. Values of the critical pressure gradient for several cases are presented.

Experimental Investigation of Pressure Drop Through Ceramic Foams: An Empirical Model for Hot and Cold Flow

2011 ◽  
Vol 133 (11) ◽  
Author(s):  
S. Akbar Shakiba ◽  
R. Ebrahimi ◽  
M. Shams
Keyword(s):  
Pressure Drop ◽  
Empirical Model ◽  
Reynolds Numbers ◽  
Dimensionless Numbers ◽  
Ceramic Foams ◽  
Cold Flow ◽  
Leading Role ◽  
Set Up ◽  
Made In

Growing application and use of ceramic foams has intensified the necessity to determine a precise and inexpensive method for prediction of pressure drop through these materials. In this paper, a new experimental model is presented for pressure drop through ceramic foams. In order to measure pressure drop, a set up was made in which air flow rate and temperature varied. Effects of variation in temperature and flow velocity on the pressure drop were investigated through open-cell SiC and Al2O3 foams with different values of porosity and pore density. Results of this study revealed the leading role of parameters such as viscosity, porosity, density, velocity and mean hydraulic diameter of pores of foam. Since there are several parameters affecting the problem, dimensional analysis was adopted as a convenient approach. Euler number, porosity and two Reynolds numbers, one based on the pores’ diameter and the other one based on total bed length, have been shown to be important in the analysis. Finally, an empirical model is developed for the pressure drop which is based on dimensionless numbers.

scholarly journals Prediction and Measurement of Pressure Drop of Water Flowing in a Rectangular Microchannel

2013 ◽  
Vol 3 (2) ◽  
Author(s):  
M. Mirmanto
Keyword(s):  
Experimental Data ◽  
Reynolds Number ◽  
Pressure Drop ◽  
Reynolds Numbers ◽  
Hydraulic Diameter ◽  
Inlet Temperature ◽  
Mean Velocity ◽  
Rectangular Microchannel ◽  
The Mean ◽  
Poiseuille Number

This paper presents experimental results of pressure drop measurement and prediction of water flowing through a copper rectangular microchannel with a hydraulic diameter of 437 µm. The aim of this work is to identify discrepancies between experimental data and macrochannel theory. An inlet temperature of 60oC was kept constant at the channel entrance and the experiments were performed with Reynolds numbers (based on the mean velocity and hydraulic diameter) ranging up to 4500.  The results show that the pressure drop prediction agrees with the theory. However, the trend of Poiseuille number with the Reynolds number was not constant for laminar flow. This could be due to the entrance effect. Moreover, the friction factor theory could predict the experimental data for turbulent flow. Thus, in this experiment, the theory for flow in macro passages is still applicable.

POTRET PERILAKU RELIGIUS MAHASISWA UIN SAYARIF HIDAYATULLAH JAKARTA

Al-Risalah ◽  
2018 ◽  
Vol 8 (2) ◽  
pp. 126-148
Author(s):  
Ay Maryani
Keyword(s):  
Religious Behavior ◽  
Male And Female ◽  
Mean Values ◽  
Significance Level ◽  
Average Value ◽  
Environment Variable ◽  
Independent Variable ◽  
The Mean ◽  
Gender Factor ◽  
Level 2

This study describes the religious behavior of UIN Syarif Hidayatullah Jakarta students. The variables used are internal and external environment as independent variable and religious behavior as dependent variable. The internal environment variable consists of gender factor, (male and female). The external environmental variables comprise the faculty environment, the present resident and the place to grow. The present resident consist of dormitories, boarding houses and parents' homes and the place to grow consist of urban and rural environment. Religious behavior variables consist of habluminallah behavior and habluminannas behavior. Habluminallah's behavior measured by (1) knowledge of faith and worship, (2) attitudes toward faith and worship, and (3) practice of faith and worship. The habluminannas variable measured by (1) Islamic behavior for them self, like honest, discipline and good work / studyethics, trust and concern on legality, (2) Islamic behavior with others, like generous, cooperation, caring, respect to the people's rights and tolerance and (3) Islamic behavior for the natural surroundings, like love of nature and nature conservation efforts. The methodology used was (1) Statistical descriptive, (2) MANOVA (Multivariate Analysis of Variance) and (3) Independent sample t-test. The results showed the religious behavior UIN Syarif Hidayatullah Jakarta students has a very good category. This is indicated by the mean values for the behavior of haluminallah and habluminannas of 158.85 and 178.76,  espectively. The average value of habluminallah behaviour in the range of values "145-180" with the category of "very good" and habluminannas behavior in the range value "165-205" with the category "very good. Habluminallah and habluminannas behavior are different for each faculty. This is indicated by the significance level of Pillai Trace, Wilk Lambda, Hotelling Trace, Roy's Largest Root of 0.00 (<0.05). Habluminallah and habluminannas behavior are the same for respondents who live in dormitories, boarding houses, and parents' homes. This is indicated by the value of F test and significance at Wilk's Lambda respectively for 2.055 and 0.085 (>0.05). Habluminallah and habluminannas behaviors are similar for urban and rural respondents. This is known from the sig level. (2-tailed) for habluminallah and habluminannas behavior of 0.317 and 0.245 (> 0.05), respectively. Habluminallah and habluminannas behaviors are similar for male and female. This is known from the sig level. (2-tailed) for habluminallah and habluminannas behavior of 0.950 and 0.307 (> 0.05),respectively. The results of this study are expected to be used to develop university policies that can enhance the Islamic values of UIN Syarif Hidayatullah Jakarta.

scholarly journals Use and success evaluation of percutaneous aortic balloon valvuloplasty in different hemodynamic entities of severe aortic stenosis in the TAVR era

2020 ◽  
Vol 41 (Supplement_2) ◽  
Author(s):  
K Piayda ◽  
A Wimmer ◽  
H Sievert ◽  
K Hellhammer ◽  
S Afzal ◽  
...  
Keyword(s):  
Heart Failure ◽  
Aortic Valve ◽  
Aortic Stenosis ◽  
Aortic Valve Replacement ◽  
Pressure Gradient ◽  
Severe Aortic Stenosis ◽  
Primary Treatment ◽  
Low Flow ◽  
Mean Pressure ◽  
The Mean

Abstract Background In the era of transcatheter aortic valve replacement (TAVR), there is renewed interest in percutaneous balloon aortic valvuloplasty (BAV), which may qualify as the primary treatment option of choice in special clinical situations. Success of BAV is commonly defined as a significant mean pressure gradient reduction after the procedure. Purpose To evaluate the correlation of the mean pressure gradient reduction and increase in the aortic valve area (AVA) in different flow and gradient patterns of severe aortic stenosis (AS). Methods Consecutive patients from 01/2010 to 03/2018 undergoing BAV were divided into normal-flow high-gradient (NFHG), low-flow low-gradient (LFLG) and paradoxical low-flow low-gradient (pLFLG) AS. Baseline characteristics, hemodynamic and clinical information were collected and compared. Additionally, the clinical pathway of patients (BAV as a stand-alone procedure or BAV as a bridge to aortic valve replacement) was followed-up. Results One-hundred-fifty-six patients were grouped into NFHG (n=68, 43.5%), LFLG (n=68, 43.5%) and pLFLG (n=20, 12.8%) AS. Underlying reasons for BAV and not TAVR/SAVR as the primary treatment option are displayed in Figure 1. Spearman correlation revealed that the mean pressure gradient reduction had a moderate correlation with the increase in the AVA in patients with NFHG AS (r: 0.529, p&lt;0.001) but showed no association in patients with LFLG (r: 0.145, p=0.239) and pLFLG (r: 0.030, p=0.889) AS. Underlying reasons for patients to undergo BAV and not TAVR/SAVR varied between groups, however cardiogenic shock or refractory heart failure (overall 46.8%) were the most common ones. After the procedure, independent of the hemodynamic AS entity, patients showed a functional improvement, represented by substantially lower NYHA class levels (p&lt;0.001), lower NT-pro BNP levels (p=0.003) and a numerical but non-significant improvement in other echocardiographic parameters like the left ventricular ejection fraction (p=0.163) and tricuspid annular plane systolic excursion (TAPSE, p=0.066). An unplanned cardiac re-admission due to heart failure was necessary in 23.7% patients. Less than half of the patients (44.2%) received BAV as a bridge to TAVR/SAVR (median time to bridge 64 days). Survival was significantly increased in patients having BAV as a staged procedure (log-rank p&lt;0.001). Conclusion In daily clinical practice, the mean pressure gradient reduction might be an adequate surrogate of BAV success in patients with NFHG AS but is not suitable for patients with other hemodynamic entities of AS. In those patients, TTE should be directly performed in the catheter laboratory to correctly assess the increase of the AVA. BAV as a staged procedure in selected clinical scenarios increases survival and is a considerable option in all flow states of severe AS. (NCT04053192) Figure 1 Funding Acknowledgement Type of funding source: None

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