Reynolds Transport Theorem Steven A Jones Biomedical Engineering

Reynolds Transport Theorem Steven A. Jones Biomedical Engineering January 8, 2008 Louisiana Tech University Ruston, LA 71272

Things to File Away • Divergence Theorem • If the integral of some differential entity over an arbitrary sample volume is zero, then the differential entity itself is zero. Louisiana Tech University Ruston, LA 71272

Conservation Laws • Conservation of mass: Increase of mass = mass generated + mass flux • Conservation of momentum Increase of momentum = momentum generated + momentum flux • Conservation of energy Increase of energy = energy generated + energy flux If more mass goes in than comes out, mass accumulates (unless it is destroyed). If we take in more calories than we use, we get fat. Louisiana Tech University Ruston, LA 71272

Conservation Laws: Mathematically All three conservation laws can be expressed mathematically as follows: Production of the entity (e. g. mass, momentum, energy) Increase of “entity per unit volume” Flux of “entity per unit volume” out of the surface of the volume (n is the outward normal) is some entity. It could be mass, energy or momentum. is some property per unit volume. It could be density, or specific energy, or momentum per unit volume. Louisiana Tech University Ruston, LA 71272

The Bowling Ball If you are on a skateboard, traveling west and someone throws a bowling ball to you from the south, what happens to your westward velocity component? (You slow down). Louisiana Tech University Ruston, LA 71272

Reynolds Transport Theorem: Mass If we are concerned with the entity “mass, ” then the “property” is mass per unit volume, i. e. density. Production of mass within the volume Effect of increased mass on density within the volume. Flux of mass through the surface of the volume Mass can be produced by: 1. Nuclear reactions. 2. Considering a certain species (e. g. production of ATP). Louisiana Tech University Ruston, LA 71272

Reynolds Transport Theorem: Momentum If we are concerned with the entity “mass, ” then the “property” is mass per unit volume, i. e. density. Production of momentum within the volume Increase of momentum within the volume. Momentum can be produced by: External Forces. Louisiana Tech University Ruston, LA 71272 Flux of momentum through the surface of the volume

Mass Conservation in an Alveolus Density remains constant, but mass increases because the control volume (the alveolus) increases in size. Thus, the limits of the integration change with time. Term 1: There is no production of mass. Control Volume (CV) Term 2: Density is constant, but the control volume is growing in time, so this term is positive. Control Surface CS Term 3: Flow of air is into the alveolus at the inlet, so this term is negative and cancels Term 2. Louisiana Tech University Ruston, LA 71272

Mass Conservation in an Alveolus N 2 , O 2 , CO 2 and others. O 2 Can look separately at O 2 and CO 2 Third term is different: (Inflow of O 2 from the bronchiole) – (Outflow of O 2 into the capillary system) Louisiana Tech University Ruston, LA 71272

Heating of a Closed Alveolus Density can be “destroyed” through energy influx, but the transport theorem still holds. Heat Term 1 is zero. No mass is created inside the control volume. Term 2 is zero. The decrease in density is cancelled by the increase in volume. Term 3 is zero. There is no flux of mass through the walls. Louisiana Tech University Ruston, LA 71272

Air Compressed into a Rigid Vessel Density increases so mass increases while the control volume (vessel) remains constant. Term 1: There is no production of mass in the container. Term 2: There is an increase in the total mass of air in the container. Region R(m) Surface S(m) Louisiana Tech University Ruston, LA 71272 Term 3: There is flow of air into the alveolus at the inlet.

Differential Form dx Along the 2 faces shown, vy and vz do not contribute to changes in the mass within the cube. Only vx contributes. dy dz The left hand term is production of mass. The first term on the right is an increase in density within the cube, and the second term on the right is the outward flux of fluid. If the control volume is stationary, then: Because mass is not being created or destroyed, the left hand term is 0. Louisiana Tech University Ruston, LA 71272

Differential Form – Conservation of Mass dx dy dz Louisiana Tech University Ruston, LA 71272 We can get a differential form if we convert the last integral to a volume integral. The divergence theorem says:

Differential Form Continuity Equation, Differential Form Louisiana Tech University Ruston, LA 71272

Divergence Conservation of mass reduces to: If density is constant then When is density constant? Louisiana Tech University Ruston, LA 71272

Constant Density • Generally density is taken as constant when the Mach number M v/c is much less than 1 (where c is the speed of sound). • For biological and chemical applications, this condition is almost always true. • For design of aircraft, changes in density cannot necessarily be ignored. • In acoustics (but nobody pays any attention when I say this). Louisiana Tech University Ruston, LA 71272

RTT Applied to Momentum Production of the entity (e. g. mass, momentum, energy) Increase of “entity per unit volume” is now momentum. is momentum per unit volume. Louisiana Tech University Ruston, LA 71272 Flux of “entity per unit volume” out of the surface of the volume (n is the outward normal)

RTT Applied to Momentum Production of the entity momentum Increase of “momentum per unit volume” Flux of “momentum per unit volume” out of the surface of the volume (n is the outward normal) is now momentum. is momentum per unit volume. Louisiana Tech University Ruston, LA 71272

RTT Applied to Momentum This v is part of the property being transported. This v transports the property. Louisiana Tech University Ruston, LA 71272

RTT Applied to Momentum has three components. Therefore, this is really 3 equations. Momentum is “produced” by external forces. Therefore, the first term represents the forces on the control volume. Louisiana Tech University Ruston, LA 71272

Example 3. 7 v 2 v 1 White reduces to: Louisiana Tech University Ruston, LA 71272 What resultant force is required to hold the section of tubing in place? Steady state

Example 3. 7 v 2 v 1 Louisiana Tech University Ruston, LA 71272

Momentum and Pressure Pout Pin twall Louisiana Tech University Ruston, LA 71272 CS

Example 3. 1 from White 2 Find the rate of change of energy in the control volume. CV 1 3 Section 1 2 3 Type Inlet Outlet Louisiana Tech University Ruston, LA 71272 r (kg/m 2) 800 800 V (m/s) 5 8 17 A (m 2) 2 3 2 e (J/kg) 300 150

Example 3. 1 Continued If the system is in steady state (i. e. there is no change with time of the energy within the control volume), then the integral is zero. Thus, the loss of energy through the control surface must be balanced by a “production” of energy. This example is a bit misleading because “production” may be considered to be a flux of energy through the control surface. However, production could also be caused by, for example, a chemical reaction. Louisiana Tech University Ruston, LA 71272

Example 3. 1 Continued 4. 08 MW -1. 92 MW -2. 4 MW Louisiana Tech University Ruston, LA 71272 -2. 4 + -1. 92 = -4. 32 (>4. 08), so there is more energy coming in than going out. Therefore, the “box” must “destroy” the energy (e. g. by doing work).