Basic Fluid Mechanics

Transcription

1 Basic Fluid Mechanics Chapter 3B: Conservation of Mass C3B: Conservation of Mass Governing Equations There are two basic types of governing equations that we will encounter in this course Differential equations which will hold at a point in a flow field. Integral equations which hold for some volume of fluid. These governing equations are derived from the principles of Conservation of Mass Conservation of Momentum (Newton's second law) Conservation of Energy In this course we will not consider heat transfer and therefore will use the first 2 principles. C3B: Conservation of Mass 2 1

2 3.2.1 Conservation of Mass The conservation of mass (also referred to as the continuity eq) states that mass cannot be created or destroyed (neglecting nuclear effects). It can be written in three-dimensions in Differential form as: t x y z u v w 0 (3.3) or in Integral form: t C d (3.4) S V da 0 C3B: Conservation of Mass Conservation of Mass Example 2.1: Air flows in a channel whose velocity at three adjacent points A, B, and C, which are 10 cm apart, are shown in the following figure. Find the density gradient at point B if the fluid temperature is 40 degrees C and the absolute static pressure is 350 kpa. Solution: Use the differential form of the continuity equation. Assume that the flow is steady and uniform. Write an approximate eq for the change in velocity with x distance. Use continuity to solve for the density gradient. C3B: Conservation of Mass 4 2

3 3.2.1 Conservation of Mass The differential form of the continuity equation (Eq 3.3). Assume that the flow is steady, uniform, and 1D. Equation (3.3) becomes Eq A C3B: Conservation of Mass Conservation of Mass Approximate simplified Eq 3.3 Therefore, C3B: Conservation of Mass 6 3

4 3.2.1 Conservation of Mass Compute the density using the Ideal Gas Law Sub back into Eq A kg / m ====The End==== C3B: Conservation of Mass 7 C3B: Conservation of Mass 8 4

5 C3B: Conservation of Mass 9 C3B: Conservation of Mass 10 5

6 In words: The time rate of change of mass within the volume must exactly balance the sum of the mass flux entering & leaving the volume. C3B: Conservation of Mass 11 Previous examples did not differentiate between the mass flow entering and leaving the volume. Let the mass flow into the volume be simply the negative of the mass flow out of the volume. The steady flow continuity equation; Problem 3.3: Apply the integral form of the continuity equation to a steady flow through an arbitrary shaped duct C3B: Conservation of Mass 12 6

7 Solution: Draw a control volume consisting of planes 1, 2, and the duct walls and use equation (3.7a) to write; Next we note that the normal velocity across AB and DC is zero since they are solid walls. Therefore Now determine: C3B: Conservation of Mass 13 Now since plane DA is a constant x plane, the unit normal vector is in the -x direction. Therefore, nˆ DA So that Similarly Hence, ====The End==== C3B: Conservation of Mass 14 7

8 Problem 3.4: Will the computed mass flow rate be different if an alternate area is selected? Solution: To answer this concern, consider a simple tube with a steady flow through it. C3B: Conservation of Mass 15 Since u is a constant across A and the fluid is incompressible. Note: Therefore: Hence, it does not matter which area or velocity you choose as long as you are consistent! ====The End==== C3B: Conservation of Mass 16 8

9 Problem 4.1: Find the velocity at section 2. Given: Steady flow of water through the adjacent device. All properties are uniform at the ports. Solution: C3B: Conservation of Mass 17 C3B: Conservation of Mass 18 9

10 C3B: Conservation of Mass 19 ====The End==== C3B: Conservation of Mass 20 10

11 C3B: Conservation of Mass 21 C3B: Conservation of Mass 22 11

12 C3B: Conservation of Mass 23 m bc m bc indicates flow out across surface bc. V da kg / s A bc ====The End==== C3B: Conservation of Mass 24 12