Part 5: Mass, Bernoulli, and Energy Equations

Presentation Transcript

Part 5: Mass, Bernoulli, and Energy Equations Eric G. Paterson Department of Mechanical and Nuclear Engineering The Pennsylvania State University Spring 2005

Note to Instructors These slides were created 1 amid the spring semester 2005, as a showing help for the undergrad Fluid Mechanics course (ME33: Fluid Flow) in the Department of Mechanical and Nuclear Engineering at Penn State University. This course had two segments, one educated without anyone else and one instructed by Prof. John Cimbala. While we gave basic homework and exams, we autonomously created address notes. This was additionally the principal semester that Fluid Mechanics: Fundamentals and Applications was utilized at PSU. My area had 93 understudies and was held in a classroom with a PC, projector, and board. While slides have been created for every section of Fluid Mechanics: Fundamentals and Applications, I utilized a blend of writing board and electronic introduction. In the understudy assessments of my course, there were both positive and negative remarks on the utilization of electronic introduction. Along these lines, these slides ought to just be coordinated into your addresses with watchful thought of your showing style and course targets. Eric Paterson Penn State, University Park August 2005 1 These slides were initially arranged utilizing the LaTeX typesetting framework ( http://www.tug.org/) and the beamer class ( http://latex-beamer.sourceforge.net/), yet were meant PowerPoint for more extensive scattering by McGraw-Hill.

Introduction This part manages 3 conditions regularly utilized as a part of liquid mechanics The mass condition is an outflow of the preservation of mass guideline. The Bernoulli condition is worried with the preservation of active, potential, and stream energies of a liquid stream and their change to each other. The vitality condition is an announcement of the protection of vitality standard.

Objectives After finishing this section, you ought to have the capacity to Apply the mass condition to adjust the approaching and active stream rates in a stream framework. Perceive different types of mechanical vitality, and work with vitality transformation efficiencies. Comprehend the utilization and confinements of the Bernoulli condition, and apply it to understand an assortment of liquid stream issues. Work with the vitality condition communicated regarding heads, and utilize it to decide turbine control yield and pumping power necessities.

Conservation of Mass Conservation of mass standard is a standout amongst the most central standards in nature. Mass, similar to vitality, is a moderated property, and it can't be made or decimated amid a procedure. For shut frameworks mass protection is understood since the mass of the framework stays steady amid a procedure. For control volumes , mass can cross the limits which implies that we should monitor the measure of mass entering and leaving the control volume.

Mass and Volume Flow Rates The measure of mass coursing through a control surface for each unit time is known as the mass stream rate and is signified The dab over an image is utilized to show time rate of progress . Stream rate over the whole cross-sectional region of a pipe or channel is acquired by combination While this expression for is correct, it is not generally helpful for building examinations.

Average Velocity and Volume Flow Rate Integral in can be supplanted with normal estimations of r and V n For some streams variety of r is little: Volume stream rate is given by Note: numerous course books utilize Q rather than for volume stream rate. Mass and volume stream rates are connected by

Conservation of Mass Principle The preservation of mass guideline can be communicated as Where and are the aggregate rates of mass stream into and out of the CV, and dm CV/dt is the rate of progress of mass inside the CV.

Conservation of Mass Principle For CV of discretionary shape, rate of progress of mass inside the CV net mass stream rate Therefore, general protection of mass for a settled CV is:

Steady—Flow Processes For unfaltering stream, the aggregate sum of mass contained in CV is consistent. Aggregate sum of mass entering must be equivalent to aggregate sum of mass leaving For incompressible streams,

Mechanical Energy Mechanical vitality can be characterized as the type of vitality that can be changed over to mechanical work totally and specifically by a perfect mechanical gadget, for example, a perfect turbine. Stream P/r , active V 2/g , and potential gz vitality are the types of mechanical vitality e mech = P/r + V 2/g + gz Mechanical vitality change of a liquid amid incompressible stream gets to be without loses, D e mech speaks to the work provided to the liquid ( D e mech >0) or removed from the liquid ( D e mech <0).

Efficiency Transfer of e mech is generally proficient by a pivoting shaft: shaft work Pump, fan, impetus: gets shaft work (e.g., from an electric engine) and exchanges it to the liquid as mechanical vitality Turbine: changes over e mech of a liquid to shaft work. Without irreversibilities (e.g., grinding), mechanical productivity of a gadget or process can be characterized as though h mech < 100%, misfortunes have happened amid change.

Pump and Turbine Efficiencies In liquid frameworks, we are typically inspired by expanding the weight, speed, and additionally height of a liquid. In these cases, productivity is better characterized as the proportion of (provided or removed work) versus rate of increment in mechanical vitality Overall proficiency must incorporate engine or generator productivity.

General Energy Equation One of the most essential laws in nature is the first law of thermodynamics , which is otherwise called the preservation of vitality guideline . It expresses that vitality can be neither made nor annihilated amid a procedure; it can just change frames Falling rock, gets speed as PE is changed over to KE. In the event that air resistance is ignored, PE + KE = consistent

General Energy Equation The vitality substance of a shut framework can be changed by two instruments: warm exchange Q and work exchange W . Preservation of vitality for a shut framework can be communicated in rate shape as Net rate of warmth exchange to the framework: Net power contribution to the framework:

General Energy Equation Recall general RTT "Determine" vitality condition utilizing B=E and b=e Break control into rate of shaft and weight work

General Energy Equation Where does expression for weight work originate from? At the point when cylinder moves down ds affected by F=PA , the work done on the framework is d W limit =PAds. In the event that we isolate both sides by dt , we have For summed up control volumes: Note sign traditions: is outward directing ordinary Negative sign guarantees that work done is certain when is set on the framework.

General Energy Equation Moving necessary for rate of weight work to RHS of vitality condition brings about: Recall that P/r is the stream work, which is the work related with pushing a liquid into or out of a CV for each unit mass.

General Energy Equation As with the mass condition, functional examination is frequently encouraged as midpoints crosswise over deltas and ways out Since e=u+ke+pe = u+V 2/2+gz

Energy Analysis of Steady Flows For relentless stream, time rate of progress of the vitality substance of the CV is zero. This condition expresses: the net rate of vitality exchange to a CV by warmth and work exchanges amid unfaltering stream is equivalent to the contrast between the rates of active and approaching vitality streams with mass.

Energy Analysis of Steady Flows For single-stream gadgets , mass stream rate is consistent.

Energy Analysis of Steady Flows Divide by g to get each term in units of length Magnitude of each term is currently communicated as an identical segment tallness of liquid, i.e., Head

The Bernoulli Equation If we disregard channeling misfortunes, and have a framework without pumps or turbines This is the Bernoulli condition It can likewise be determined utilizing Newton's second law of movement (see content, p. 187). 3 terms relate to: Static, dynamic, and hydrostatic head (or weight).

HGL and EGL It is regularly helpful to plot mechanical vitality graphically utilizing statures. Water driven Grade Line Energy Grade Line (or aggregate vitality)

The Bernoulli Equation The Bernoulli condition is an inexact connection between weight, speed , and rise and is legitimate in areas of consistent, incompressible stream where net frictional strengths are immaterial. Condition is valuable in stream areas outside of limit layers and wakes.

The Bernoulli Equation Limitations on the utilization of the Bernoulli Equation Steady stream: d/dt = 0 Frictionless stream No pole work: w pump =w turbine =0 Incompressible stream: r = consistent No warmth exchange: q net,in =0 Applied along a streamline (with the exception of irrotational stream)