Ring Physics

Ring Physics

The Geometry and Algorithms of the Control Volume

Consider the volume between the table and the disk as a series of nesting concentric rings of radius [r], height [h] and width [dr]. This is the basic analytical control volume of the model. The model accounts for the flows into and out of each ring and also for the energy transfers within each ring

As air radiates outward, between the table and the disk, it passes through several rings

Ring Detail

Air enters the ring through the inside face and leaves through the outside face. The top and bottom faces of the ring contact the table and the disk and provide skin friction or drag.

Ring Measurements

The height [h] is the space between the disk and the table above it. The ring width [dR] is the total radius of the disk outside the central orifice divided by the number of rings chosen for the computation. The ring length [w] extends and curves around to become the circumference of the ring. Note that the distance [dR] may be quite different from the distance the gas moves during the time interval [dT]. The distance the gas moves is its velocity times the time interval [dT].

The Ring - Flows In and Out

Consider the control volume between the inner face [I] and the outer face [O]. Some of the air (green) leaves the ring and flows into the next larger ring. The trailing edge of the remaining air (yellow) moves up to point [C]. Some air from the next smaller ring (blue) flows in from the left and up to point [A]. Now consider a point [B] such that the volumes on either side of point B are proportional to the masses of the new and old gas. The diagram happens to show A < B < C although that would not necessarily have to the order.

For purposes of computation, the new air and the old air then expand adiabatically into their respective volumes. Finally, their pressures and temperatures mix to determine the total pressure and temperature.

Ring to Ring Flow Computation Algorithm

The goal is to update the mass [M], pressure [P], temperature [T] and momentum, [X] states of the control volume in response to the flows occurring in and out during a small time interval [dT]. Rather than using the standard approach (balancing the work done as the gas flows in and out of the control volume) consider the flows in and out are the result of Radial Momentum. As such, assume the gas arriving and leaving carries the of the control volume from which it originates. Assume the gases will wind up occupying volumes proportional to their masses. So, first perform and adiabatic expansion (or compression) of both gasses into these volumes. Then combine the two gasses to find the new total.

Find the starting volumes from the gas velocities and their control volumes:

Volume_old_1 = Volume_Total * (1 - Velocity_old * dT / dR)

Volume_new_1 = Volume_Total[r-1] * (Velocity[r-1] * dT/ dR)

Find the masses of the gasses from their volumes and densities:

Mass_old = Volume_old * density_old

Mass_new = Volume_new * density_new

Find the new volumes by partitioning the control according to mass:

Volume_old_2 = (O - I) * Face * M_old / (M_old + M_new)

Volume_new_2 = (O - I) * Face * M_new / (M_old + M_new)

Expand (or compress both gasses adiabiatically) into their new volumes:

Pressure_old = Pressure_old * (Volume_old_1/ Volume_old_2)^1.4

Temperature_old = T * (P/P)

Pressure_new = Pressure_new * (Volume_new_1/ Volume_new_2)^1.4

Temperature_new = = T * (P/P)

Mix the Gasses:

Mass_total = Mass_old _ Mass_new

Pressure_total = (Pressure_new * Mass_new + Pressure_old * Mass_old) / Mass_total

Temperature_total = (Temperature_new * Mass_new + Temperature_old * Mass_old) / Mass_total

Momentum is the sum of the momenta:

Momentum_total = Momentum_old + Momentum_new

Internal Ring Activity Computation Algorithm

Within a ring, the states change through drag and pressure acceleration.

Drag is the resistive pressure of the air against the table and the disk. At low Reynolds this is generally a viscosity effect based on the non-slip property. At high Reynolds number friction is generally a kinetic roughness effect. See Friction Studies for more details. Drag is proportional to the contact surface area (the sum of the top and bottom areas). The pressure gradient across the ring width [dR] times the face normal to the flow gives the resistive force. The resulting acceleration is the drag from the top and bottom faces divided by the mass in the ring. This acceleration decrements the forward velocity over the next time interval [dT]. The drag also generates internal thermal power equal to the acceleration times the momentum. Units: m/s² * kg-m/s = kg-m²/s³. This thermal power increments the thermal energy of the gas and raises the temperature.

Drag_Force = Drag_Pressure * Face_top

Drag_Acceleration = Drag_Force / Mass

Velocity = Velocity - Drag_Acceleration * dT

Drag_Power = Drag_Acceleration * Momentum

Thermal_Energy = 3/2 * Mass * k * Temperature + Drag_Power * dT

Temperature = 2/3 * Thermal_Energy / (Mass * k)

Pressure acceleration is the effect of the pressure gradient across the width of a ring to change the velocity of the gas. As with drag, the force equals the pressure times the face area normal the flow. In this case, however, the force is 1/2 the pressure drop since the average gas molecule is just half way between the inner and outer faces. The resultant acceleration increments or decrements the velocity. In addition, the acceleration attends a power loss that this model carries to debit the very pressure that induced it.

Pressure_Acceleration = Pressure_Drop / Mass / 2.0

Velocity = Velocity + Pressure_Acceleration * dT

Pressure_Power_Loss = Pressure_Acceleration * Momentum

Pressure_Energy = Pressure * Volume - Pressure_Power_Loss * dT

Pressure = Pressure_Energy / Volume

A Palimpsest of Levitator Equations

Variable	Equation	Notes
Face Area (m²)		Circumference times height
Frequency (1/s)		Velocity divided by length
Mass (kg)		Accumulate mass flux.
Mass Flux (kg/s)		Mass times frequency
Momentum (kg-m/s)		Accumulate momentum flux.
Momentum Flux (kg-m/s²)		Momentum times frequency
Pressure (pa)		Energy divided by the 1.4 power of volume
Thermal Energy (J)		Accumulate thermal energy flux
Thermal Energy Flux (J/s)		Thermal energy times frequency
Turnover (fraction of mass)		Frequency times delta time
Top Area (m²)		Circumference times width
Velocity (m/s)		Momentum divided by mass.
Volume (m³)		Circumference times width times height

Gas Physics¹ - A Review of Basic Principles of Gas Physics.

The Energy and pressure of a molecule are a function of velocity.

A molecule bouncing around inside a cube has kinetic energy, E = mv² and pressure, P = 1/3 mv²/V.

Consider a cube of side L containing a molecule of a gas with momentum mv_x. Each time it hits and recoils from a wall, it imparts an impulse, i = 2 * mv_x. It repeats this with a frequency, f = v_x/(2*L) so the force on the wall, F = f * i = mv_x²/L. For a number [N] of molecules, the force is F = Nmv_x²/L. Now since the molecule may move in any one of three directions the average velocity [v²] = 3 * v_x². Thus, the pressure on any one wall [P] = F/A = 1/3 Nmv²/V and PV = 1/3 Nmv².

Since the kinetic energy [Ek] = 1/2 Nmv², PV = 2/3 Ek and Ek = 3/2 PV.

And since PV = nRT = NkT, E = 3/2 NkT and T = 2/3 E / (Nk)

Note: R = 8.31 J/mol-K and k = R/6*10²³ = 1.38 * 10-²³ J/molecule-K

Isothermal Expansion

For isothermal (constant temperature) processes, consider PV = NkT. Since T is constant, P = NkT / V, and pressure is directly inverse to volume.

Adiabatic Expansion

For adiabatic (no heat flows in or out to the control volume) expansion, temperature is free to vary. Temperature generally falls as volume rises and as pressure falls. In particular, PV^g = K where g = cp/cv. For diatomic gasses such as nitrogen and oxygen g = 1.4. Therefore P = P₀*(V₀ / V)^1.4 and pressure drops a little faster than it would if the gas were free to absorb heat and maintain constant temperature.

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References: 1. Teaching Introductory Physics, Swartz, 1998, AIP Press, pages 236 and 239.