1. Fluid Mechanics and Applications MECN 3110
Inter American University of Puerto Rico

2. Integral Relations for a Control Volume
Chapter 3
Integral Relations for a Control Volume

3. To understand the Reynolds Transport Theorem. To apply
Course Objectives
To define volume flow rate, weight flow rate, and mass flow rate and their units.
To understand the Reynolds Transport Theorem.
To apply
Conservation of Mass Equation
Linear Momentum Equation
Energy Equation
Frictionless Flow: The Bernoulli Equation
Thermal Systems Design Universidad del Turabo

4. Introduction
All the laws of mechanics are written for a system, which is defined as an arbitrary quantity of mass of fixed identity. Everything external to this system is denoted by the term surrounding, and the system is separated fro its surrounding by its boundaries.
A control volume is defined as a specific region in the space for study.
System
Control Volume

5. Volume and Mass Rate of Flow
All the analyses in this chapter involve evaluation of the volume flow Q or mass flow m passing through a surface (imaginary) defined in the flow.

6. Volume and Mass Rate of Flow
The integral dV /dt is the total volume rate of flow Q through the surface S.
Volume flow can be multiplied by density to obtain the mass flow m. If density varies over the surface, it must be part of the surface integral
If density is constant, it comes out of the integral and a direct proportionality results:

7. Volume and Mass Rate of Flow
The quantity of fluid flowing in a system per unit time can be expressed by the following three different terms:
The volume flow rate is the volume of fluid flowing past a section per unit time
where A is the area of the section and ν is the average velocity of flow
The weight flow rate is the weight of fluid flowing past a section per unit time
where ɣ is the specific weight

8. Volume and Mass Rate of Flow
The mass flow rate is the mass of fluid flowing through a section per unit time
where ρ is the density

9. The Reynolds Transport Theorem
To convert a system analysis to a control-volume analysis is needed the Reynolds transport theorem.
Arbitrary Fixed Control Volume
Fixed Control Volume
B is any property of the fluid and β is an intensive property
Compact form of the Reynolds Transport Theorem CS

10. Application Problems

14. Conservation of Mass
For conservation of mass B is m (mass) and β is 1.
If the volume control has only a number of the one-dimensional inlets and outlets, we can write

15. Conservation of Mass
Other special cases occur. Suppose that flow within the control volume is steady, then
This states that in steady flow the mass floes entering and leaving the control volume must balance exactly. For steady flow

16. Conservation of Mass
The quantity ρVA is called mass flow m with units of kg/s or slugs/s
In general, the steady-flow mass conservation relation can be written as

17. Conservation of Mass
Incompressible Flow: The variation of density can be considered negligible.
If the inlets and outlet are one-dimensional, we have
Where Q=VA is called the volume flow passing through the given cross section.

18. Application Problems

19. Problem

20. Solution

21. Problem

22. Problem

23. Problem

24. The Linear Momentum Equation
For linear momentum equation for a deformable control-volume.
For a fixed control-volume, the relative velocity Vr=V
If the volume control has only a number of the one-dimensional inlets and outlets, we can write

25. Application Problems

26. Problem

27. Solution If
The components x and z of the linear momentum equation are:

28. Solution
Writing the previous equations in the scalar form:
Using the conservation of mass V1A1=V2A2 or A1=A2, since V1=V2.

29. Solution
Replacing the values:

30. Problem

31. Solution

32. Solution

33. Energy Equation
As the final basic law, we apply the Reynolds transport theorem to the first law of thermodynamics. The dummy variable B becomes energy E, and the energy per unit mass is β=dE/dm=e.
Positive Q denotes heat added to the system and positive W denotes work done by the system

34. Energy Equation
The Steady Flow Energy Equation If

35. Energy Equation
The Steady Flow Energy Equation
Where hf the friction loss is always positive, the pump always add energy (increase the left-hand side) hpump and the turbine extracts energy from the flow hturbine.

36. Application Problems

37. Problem

38. Problem

39. Problem

40. Problem

41. Problem

42. Frictionless Flow: The Bernoulli Equation
Closely to the steady flow energy equation is a relation between pressure, velocity, and elevation in a frictionless flow, now called the Bernoulli Equation.
For an unsteady frictionless flow
For steady frictionless flow

43. Application Problems

44. Problem

45. Problem

46. Problem

47. Problem

48. Problem

49. Problem

50. Problem