Momentum balance

From Chemepedia

The momentum balance is basically a relation between the velocity of the fluid and the mass of the fluid. The momentum balance equation is basically the statement of the second law of Newton[1].

Linear Momentum

The conservation of linear momentum states that the acceleration (or change of momentum per unit time) equals the sum of forces acting on the control volume. Since momentum is a vector quantity, each term in the balance must be a vector

            (1)

For a control volume with one well-defined exit (denoted by subscript 2) and one well-defined entrance (denoted by subscript 1), the momentum balance becomes

            (2)

where is a unit normal that points in the flow direction.

The conservation of Linear Momentum over a Control Volume helps predict forces on bends and different fittings in a piping system.

Since momentum is a vector quantity, most Control Volumes require a momentum balance in an x-direction and a y-direction. The idea builds upon the fundamental concept of vector components. The forces acting on the Control Volume Shall be broken down in to x and y components using the right trigonometric functions. As a result, one would often end up having to solve two equations in order to compute all the information from a Momentum Balance.

Angular momentum

The angular momentum is dependent on the angular velocity and the moment of inertia of the object as it is equal to the product of the angular velocity and the moment of inertia. The mathematical expression is given below as follows [2]:


            (3)


Where,

  • L = Angular momentum
  • I = Moment of inertia
  • ω = Angular Velocity

The conservation of angular momentum states that rate of change of angular momentum equals the sum of torques acting on the control volume. Since torque is a vector quantity, each term in the balance must be a vector


            (4)

References

  1. [Jones, S. A. (2006, March 17). Momentum Balance. Retrieved April 24, 2019, from http://www2.latech.edu/~sajones/Biotransport/Lecture%206a%20on%20Momentum%20Balance.ppt
  2. [Angular Momentum of a Particle. (n.d.). Retrieved April 24, 2019, from Angular Momentum: http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html