Using Bernoulli equation to solve the problem of fluid accelerating motion

For the analysis of unsteady heat conduction in solid bodies comprising heat exchange by forced convection to nearby fluids, the two feasible models are (1) the differential or distributed model and (2) the lumped capacitance model. In the latter model, the suited lumped heat equation is linear, separable, and solvable in exact, analytic form. The linear lumped heat equation is constrained by the lumped Biot number criterion Bil=h¯(V/S)/ks < 0.1, where the mean convective coefficient h¯ is affected by the imposed fluid velocity. Conversely, when the heat exchange happens by natural convection, the pertinent lumped heat equation turns nonlinear because the mean convective coefficient h¯ depends on the instantaneous mean temperature in the solid body. Undoubtedly, the nonlinear lumped heat equation must be solved with a numerical procedure, such as the classical Runge–Kutta method. Also, due to the variable mean convective coefficient h¯ (T), the lumped Biot number criterion Bil=h¯(V/S)/ks < 0.1 needs to be adjusted to Bil,max=h¯max(V/S)/ks < 0.1. Here, h¯max in natural convection cooling stands for the maximum mean convective coefficient at the initial temperature Tin and the initial time t = 0. Fortunately, by way of a temperature transformation, the nonlinear lumped heat equation can be homogenized and later channeled through a nonlinear Bernoulli equation, which admits an exact, analytic solution. This simple route paves the way to an exact, analytic mean temperature distribution T(t) applicable to a class of regular solid bodies: vertical plate, vertical cylinder, horizontal cylinder, and sphere; all solid bodies constricted by the modified lumped Biot number criterion Bil,max<0.1.

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Doppler echo evaluation of pulmonary venous-left atrial pressure gradients: human and numerical model studies

AJP Heart and Circulatory Physiology ◽

10.1152/ajpheart.2000.279.2.h594 ◽

2000 ◽

Vol 279 (2) ◽

pp. H594-H600 ◽

Cited By ~ 10

Author(s):

Michael S. Firstenberg ◽

Neil L. Greenberg ◽

Nicholas G. Smedira ◽

David L. Prior ◽

Gregory M. Scalia ◽

...

Keyword(s):

Numerical Modeling ◽

Left Atrial ◽

Atrial Pressure ◽

Left Atrial Pressure ◽

High Fidelity ◽

Pressure Gradients ◽

Bernoulli Equation ◽

Venous Flow ◽

Model Studies ◽

Pulmonary Venous Flow

The simplified Bernoulli equation relates fluid convective energy derived from flow velocities to a pressure gradient and is commonly used in clinical echocardiography to determine pressure differences across stenotic orifices. Its application to pulmonary venous flow has not been described in humans. Twelve patients undergoing cardiac surgery had simultaneous high-fidelity pulmonary venous and left atrial pressure measurements and pulmonary venous pulsed Doppler echocardiography performed. Convective gradients for the systolic (S), diastolic (D), and atrial reversal (AR) phases of pulmonary venous flow were determined using the simplified Bernoulli equation and correlated with measured actual pressure differences. A linear relationship was observed between the convective ( y) and actual ( x) pressure differences for the S ( y = 0.23 x + 0.0074, r = 0.82) and D ( y = 0.22 x + 0.092, r = 0.81) waves, but not for the AR wave ( y = 0.030 x + 0.13, r = 0.10). Numerical modeling resulted in similar slopes for the S ( y = 0.200 x − 0.127, r = 0.97), D ( y = 0.247 x − 0.354, r= 0.99), and AR ( y = 0.087 x − 0.083, r = 0.96) waves. Consistent with numerical modeling, the convective term strongly correlates with but significantly underestimates actual gradient because of large inertial forces.

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Is the use of modified Bernoulli equation valid in estimating pulmonary artery pressures in patients with severely dilated tricuspid annulus?

Journal of the American College of Cardiology ◽

10.1016/s0735-1097(02)81906-4 ◽

2002 ◽

Vol 39 ◽

pp. 425 ◽

Cited By ~ 2

Author(s):

H.Mehrdad Sadeghi ◽

Alborz Hassankhani ◽

Rod Serry ◽

Ajit Raisinghani ◽

Anthony N. DeMaria

Keyword(s):

Pulmonary Artery ◽

Bernoulli Equation ◽

Tricuspid Annulus

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The Analysis of Automotive Door Seal Energy Consumption

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.80-81.714 ◽

2011 ◽

Vol 80-81 ◽

pp. 714-718

Author(s):

Yun Kai Gao ◽

Da Wei Gao

Keyword(s):

Energy Consumption ◽

Elastic Force ◽

Bernoulli Equation ◽

Analysis Model ◽

Damping Force ◽

Total Energy Consumption ◽

Linear Elastic ◽

Non Linear ◽

Linear Damping ◽

Simplified Analysis

The seal deformation of automotive door is caused by the door compression forces, including non-linear elastic force and non-linear damping force. The working principles of them are analyzed and a new simplified analysis model is built. Based on the Bernoulli equation and the law of conservation of mass, the mathematical models are established to calculate energy consumption of the seal system. According to the analysis results, the energy consumption of non-linear elastic force and non-linear damping force are respectively 84% and 16% of the total energy consumption of the seal system. At last, the calculation data is compared with the test data and the error is less than 5%, so the calculation method proposed in this paper is observed to be accurate.

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