Rotating Furnace - Mathemetical Model

Mathemetical Model

In fluid mechanics, the state when no part of the fluid has motion relative to any other part of the fluid is called 'solid body rotation'. When the liquid has reached a state of solid body rotation, then the dynamic equilibrium can be understood as a balance of two energies: gravitational potential energy, and rotational kinetic energy. When a fluid is in solid body rotation it is the lowest state of energy that is available, because in a state of solid body rotation there is no friction to dissipate any of the energy.

In an inertial reference frame, the dynamic equilibrium cannot be understood in terms of an equilibrium of forces. This is because when the liquid is rotating, there is an unbalanced force acting on the liquid – the force of gravity is acting in a vertical direction on the liquid, and the surface of the parabolic dish exerts a normal force on the liquid resting on it. The resultant force is a net centripetal force toward the axis of rotation.

The kinetic energy of a parcel of liquid given by the formula:

In the case of circular motion the relation holds ( is in radians per second), hence

The gravitational potential energy is given by where is the acceleration of gravity and is the height of the liquid's surface above some arbitrary elevation, for instance, we can set to be the lowest liquid surface. We set the potential energy equal to the kinetic energy to find the liquid's shape: This is of the form, where is a constant, which is, by definition, a paraboloid.

It is not necessary to invoke the equality of rotational kinetic energy and gravitational potential energy. With reference to the force diagram above, the vertical component of the normal force (green arrow) must equal the weight of the parcel (red arrow), which is, and the horizontal component of the normal force must equal the centripetal force (blue arrow) that keeps the parcel in circular motion, which is . Since the green arrow is perpendicular to the surface of the liquid, the slope of the surface must equal the quotient of these forces: Cancelling the 's, integrating, and setting when leads to which is identical to the result obtained by the previous method, and likewise shows that the liquid surface is paraboloidal.

The focal length of the paraboloid is related to the angular speed at which the liquid is rotated by the equation:, where is the focal length, is the rotation speed, and is the acceleration due to gravity. They must be in compatible units so, for example, can be in metres, in radians per second, and in metres per second-squared. The angle unit in must be radians. 1 radian per second is about 9.55 rotations per minute (RPM). On the Earth's surface, is about 9.81 metres per second-squared. Putting these numbers into the equation produces the approximation:, where is the focal length in metres, and is the rotation speed in RPM.