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

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

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

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

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.

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:

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

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

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

Application Problems

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

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

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

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.

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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

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

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

Solution Replacing the values:

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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

Energy Equation The Steady Flow Energy Equation If

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.

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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

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