Chapter 6 MOMENTUM ANALYSIS OF FLOW SYSTEMS
Newton’s Laws and Conservation of Momentum 6-1C Newton’s first law states that “a body at rest remains at rest, and a body in motion remains in motion at the same velocity in a straight path when the net force acting on it is zero.” Therefore, a body tends to preserve its state or inertia. Newton’s second law states that “the acceleration of a body is proportional to the net force acting on it and is inversely proportional to its mass.” Newton’s third law states “when a body exerts a force on a second body, the second body exerts an equal and opposite force on the first.” r 6-2C Since momentum ( mV ) is the product of a vector (velocity) and a scalar (mass), momentum must be a vector that points in the same direction as the velocity vector.
6-3C The conservation of momentum principle is expressed as “the momentum of a system remains constant when the net force acting on it is zero, and thus the momentum of such systems is conserved”. The momentum of a body remains constant if the net force acting on it is zero. 6-4C Newton’s second law of motion, also called the angular momentum equation, is expressed as “the rate of change of the angular momentum of a body is equal to the net torque acting it.” For a non-rigid body with zero net torque, the angular momentum remains constant, but the angular velocity changes in accordance with Iω = constant where I is the moment of inertia of the body. 6-5C No. Two rigid bodies having the same mass and angular speed will have different angular momentums unless they also have the same moment of inertia I. Linear Momentum Equation 6-6C The relationship between the time rates of change of an extensive property for a system and for a control volume is expressed by the Reynolds transport theorem, which provides the link between the r system and control volume concepts. The linear momentum equation is obtained by setting b = V and thus r B = mV in the Reynolds transport theorem. 6-7C The forces acting on the control volume consist of body forces that act throughout the entire body of the control volume (such as gravity, electric, and magnetic forces) and surface forces that act on the control surface (such as the pressure forces and reaction forces at points of contact). The net force acting on a control volume is the sum of all body and surface forces. Fluid weight is a body force, and pressure is a surface force (acting per unit area). 6-8C All of these surface forces arise as the control volume is isolated from its surroundings for analysis, and the effect of any detached object is accounted for by a force at that location. We can minimize the number of surface forces exposed by choosing the control volume such that the forces that we are not interested in remain internal, and thus they do not complicate the analysis. A well-chosen control volume exposes only the forces that are to be determined (such as reaction forces) and a minimum number of other forces. 6-9C The momentum-flux correction factor β enables us to express the momentum flux in terms of the r r r r & ρV (V ⋅ n )dAc = β mV avg . The value of β is unity for uniform mass flow rate and mean flow velocity as
flow, such as a jet flow, nearly unity for turbulent flow (between 1.01 and 1.04), but about 1.3 for laminar flow. So it should be considered in laminar flow.
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Chapter 6 Momentum Analysis of Flow Systems 6-10C The momentum equation for steady one-dimensional flow for the case of no external forces is r r r & & F= βmV − β mV
where the left hand side is the net force acting on the control volume, and first term on the right hand side is the incoming momentum flux and...