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- Control-Volume Analysis of Mass,Momentum and Energy Study Notes for Mechanical Engineering
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Control-Volume Analysis of Mass,Momentum and Energy Study Notes for Mechanical Engineering
Velocity, pressure, temperature, density, and viscosity are the main properties that should be considered simultaneously when conducting a fluid flow examination.
In accordance with the physical phenomena such as combustion, multiphase flow, turbulence, mass transport, etc. Thermo-fluid incidents directed by governing equations are based on the laws of conservation.
Although some sources specify the expression of Navier-Stokes equations merely for the conservation of momentum, some of them also use all equations of conservation of the physical properties. Regarding the flow conditions, the N-S equations are rearranged to provide affirmative solutions in which the complexity of the problem either increases or decreases.
For instance, having a numerical case of turbulence according to the pre-calculated Reynolds number requires an appropriate turbulent model to be applied to obtain credible results.
Venant had carried out studies to explore the mathematical model of fluid flow, they had overlooked the viscous frictional force. The observation method of fluid flow based on kinematic properties is a fundamental issue for generating a convenient mathematical model. Lagrangian description of fluid motion is based on monitoring a fluid particle that is large enough to detect properties. In the Eulerian method, any specific particle across the path is not followed; instead, the velocity field as a function of time and position is examined.
The missile example Figure 3 precisely fits to emphasize these methods. The Lagrangian formulation of motion is always time-dependent. Description of motion for Lagrangian:.
The equations of conservation in the Eulerian system in which fluid motion is described are expressed as Continuity Equation for mass, Navier-Stokes Equations for momentum and Energy Equation for the first law of Thermodynamics. The equations are all considered simultaneously to examine fluid and flow fields.
The mass in the control volume can be neither be created nor destroyed. The conservation of mass states that the mass flow difference throughout the system between inlet and outlet is zero:.
While the density is constant, the fluid is assumed incompressible and then continuity is simplified as below, which indicates a steady-state process:. In case of a fluid, it is convenient to express the equation in terms of the volume of the particle as follows:.
Substitution of equation 10 into 11 results in the Navier-Stokes equations of Newtonian viscous fluid in one equation:. Equation 12 is convenient for fluid and flow fields which are both transient and compressible. Thus, the Navier-Stokes equations for an incompressible three-dimensional flow can be expressed as follows:.
Besides, the energy equation has to be considered if any thermal interaction is available in the problem. Conservation of Energy is the first law of thermodynamics which states that the sum of the work and heat added to the system will result in the increase of the total energy of the system:.
One of the common types of energy equation is:. The Navier-Stokes equations have a non-linear structure with various complexities and thus it is hardly possible to conduct an exact solution for those equations. Consequently, different assumptions are required to grind the equations to a possible solution. The mathematical model merely gives ties among parameters that are part of the whole process.
Hence, the solution of the Navier-Stokes equations can be realized with either analytical or numerical methods. The analytical method only compensates solutions in which non-linear and complex structures in the Navier-Stokes equations are ignored within several assumptions. On the other hand, almost every case in fluid dynamics comprises non-linear and complex structures in the mathematical model which cannot be ignored.
A step by step computational analysis of fluid flow can be described as shown in Figure 4. The analysis of fluid flow can be conducted in either steady time-independent or unsteady time-dependent conditions. While the steady flow assumption negates the effect of some non-linear terms and provides a convenient solution, the variation of density is still a hurdle that keeps the equation in a complex formation.
Due to the malleable structure of fluids, the compressibility of particles is a significant issue. Despite the fact that all types of fluid flow are compressible in a various range of molecular structure, most of them can be assumed incompressible in which the density changes are negligible.
Having said that, high speed flows where the velocity is beyond a critical limit cannot be assumed incompressible. When the Mach number is lower than 0. The Reynolds number, the ratio of inertial and viscous effects, is also effective on Navier-Stokes equations to truncate the mathematical model. Even though viscous effects are relatively important for fluids, the inviscid flow model partially provides a reliable mathematical model to predict real process for some specific cases.
For instance, high-speed external flow over bodies is a broadly used approximation where the inviscid approach reasonably fits.
Having tangible viscous effects, creeping flow is a suitable approach to investigate the flow of lava, swimming of microorganisms, flow of polymers, lubrication, etc. The behavior of the fluid under dynamic conditions can be classified as laminar and turbulent. The laminar flow is orderly in which the motion of a fluid can be predicted precisely.
The turbulent flow has a chaotic behavior and therefore it is hard to predict the fluid flow which shows a chaotic behavior. The Reynolds number predicts the behavior of fluid flow whether laminar or turbulent regarding several properties such as velocity, length, viscosity, and also type of flow. Whilst the flow is turbulent, a proper mathematical model is selected to carry out numerical solutions.
Various turbulent models are available in the literature and each of them has a slightly different structure to examine chaotic fluid flow. Turbulent flow can be applied to the Navier-Stokes equations to model the chaotic behavior. Apart from the laminar, transport quantities of the turbulent flow, it is driven by instantaneous values. Direct numerical simulation DNS is the approach to solving the N-S equations with instantaneous values. Having distinct fluctuations varying in a broad range, DNS needs enormous effort and expensive computational facilities.
To avoid those hurdles, the instantaneous quantities are reinstated by the sum of their mean and fluctuating parts as follows:. The differences among values are shown in Figure 5 both for steady and unsteady conditions:. The fluctuations can be negligible for most engineering cases.
Thus, the RANS turbulence model is a procedure to close the system of mean flow equations. Likewise, Large Eddy Simulation LES is another mathematical method for turbulent flows which is also comprehensively applied for several cases. The simple form of the Navier-Stokes equations only encompasses the change in properties such as velocity, pressure, and density under dynamic conditions for one phase laminar flow.
Most engineering applications require further mathematical models for numerical simulation. Some common engineering problems and their relevant mathematical models are given as below:. The Navier-Stokes equations cannot compensate for the physical model of the flow at very small scales such as the motion of single bacteria — also called microfluidics.
Thereby, it is convenient to either change or reinstate the Navier-Stokes model with a suitable mathematical model. The Knudsen number Kn is a dimensionless number that is the ratio of the mean free path of molecular structure to the observation scale.
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Figure 3: Observation of fluid motion with the methods Lagrangian and Euler. In lagrangian approach the man is steady with respect to the missile which is otherwise in the Eulerian approach. References White, Frank Viscous Fluid Flow. McGraw-Hill Mechanical Engineering. ISBN Stokes, George Transactions of the Cambridge Philosophical Society. White, Frank Fluid Mechanics.
McGraw-Hill Higher Education. ISBN: Cebeci, T. Computational Fluid Dynamics for Engineers. Horizon Publishing Inc. Transport Phenomena, 2th edition.
In continuum mechanics and thermodynamics , a control volume is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference , it is a fictitious volume fixed in space or moving with constant flow velocity through which the continuum gas , liquid or solid flows. The surface enclosing the control volume is referred to as the control surface. At steady state , a control volume can be thought of as an arbitrary volume in which the mass of the continuum remains constant. As a continuum moves through the control volume, the mass entering the control volume is equal to the mass leaving the control volume.
This chapter presents derivations of the differential equations that, with corresponding boundary conditions, describe convective heat transfer processes. Since convective heat transfer always involves transfer of mass and momentum, the derivations of the corresponding equations are also presented and serve as an introduction to the heat-transfer equations, which are conceptually rather similar. The derivations in Sections 2. These derivations use control-volume analysis, together with the laws for heat- and momentum-flux rates in a viscous conducting fluid that were introduced in Chapter 1. The equations for three-dimensional flows contain extra terms but no new principles see Problems 2. Since most practical cases of convective heat transfer involve turbulent flow, the usual decomposition of the velocity and fluid properties into mean and fluctuating quantities, with subsequent averaging of the equations, is described in Section 2. Unable to display preview.
Control-Volume Analysis of Mass,Momentum and Energy Study Notes for Mechanical Engineering · A fixed mass of a fluid element in the flow-.
Control volume analysis of mass momentum and energy pdf
A fluid dynamic system can be analyzed using a control volume , which is an imaginary surface enclosing a volume of interest. The control volume can be fixed or moving, and it can be rigid or deformable. Thus, we will have to write the most general case of the laws of mechanics to deal with control volumes.
It is based on the law of conservation of momentum or on the momentum principle, which states that the net force acting on a fluid mass is equal to the change in momentum of flow per unit time in that direction. The impulse-momentum equation 1 is used to determine the resultant force exerted by a flowing fluid on a pipe bend. Let v 1 , p 1 and A 1 are the velocity, pressure and area at the section 1 of the pipe. Moment of momentum equation is derived from moment of momentum principle which states that the resulting torque acting on a rotating fluid is equal to the rate of change of moment of momentum.
CEE April 10, School of Civil and Monroe L. Conservation of Mass. The reservoir surface is 5 m x 5 m. How fast is the reservoir surface h dropping?
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