First Law for A Control Volume

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First Law for A Control Volume

• P M V Subbarao
• Professor
• Mechanical Engineering Department

Modeling of True Engineering Systems..
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Laws of Nature for A Control Mass
Conservation of Mass
Conservation of Momentum
First law of thermodynamics
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Rate Equations for Laws of Nature for A Control
Mass
Conservation of Mass
Conservation of Momentum
First law of thermodynamics
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The Family of Thermodynamic Systems
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Engineering Symptoms of Civilization
The Pinnacle of Civilization
The Onset of Civilization
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An Important Innovation
Development of Reactors
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CVs for Day to Day Use
Supply Use of LPG through Cylinders
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Domestic Using of LPG
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Control Mass or Control Volume

• A Representation for Engineering Convenience

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CM CV Representation of a Device

• Control Mass representation of Can
• The total Deodorant
• Control Volume representation of Can
• Deodorant in side the can
• At time t 0, (before spray).
• The total mass of Deodorant Mass of Deodorant
  in side the can
• Control mass is same as control volume

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CM CV Representation of a Device

• At t dt (after spraying)
• Control mass control volume spray
• It is possible to relate CM and CV of a device at
  any instant!
• Principle of Conservation mass says that the rate
  of change of mass for a control mass is always
  zero.
• What about control volume?

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The Thermodynamic Control Volume

• In real engineering devices, we are usually
  interested in a region of space, i.e, control
  volume and not particular control mass.
• The laws of nature are connected to Control Mass.
• Therefore, we need to transform Laws of
  Conservation for a control mass to a control
  volume.
• This is accomplished through the use of Reynolds
  Transport Theorem.
• Specially derived in thermodynamics for CV .

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Flowing Fluid Through A CV

• A typical control volume for flow in an
  funnel-shaped pipe is bounded by the pipe wall
  and the broken lines.
• At time t0, all the fluid (control mass) is
  inside the control volume.

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• The fluid that was in the control volume at time
  t0 will be seen at time t0 dt as           .

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The control volume at time t0dt           .


The control mass at time t0 dt        .
There will be differences between the fluid
(control mass) and the control volume at time t0
dt        .
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A Generalized Functional Model for CV

• Consider a control mass and a control volume
  (C.V.) as follows
• the control mass occupies region I and C.V.
  (region II) at time t0.
• Fluid particles of region I are trying to enter
  C.V. (II) at time t0.

III
II

• the same control mass occupies regions (IIIII)
  at t0 dt
• Fluid particles of I will enter CV-II in a time
  dt.
• Few more fluid particles which belong to CV II
  at t0 will occupy III at time t0 dt.

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Reynolds' Transport Theorem
III has left CV at time t0dt
I is trying to enter CV at time t0
The control volume may move as time passes.
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For and infinitesimal time duration
A Simple Accounting !!!
The above mentioned change has occurred over a
time dt, therefore Time averaged change in any
general property of a control mass, BCM is

• The rate of change of property B of the system.

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Conservation of Mass

• Let B mass of the system, m.

The rate of change of mass in a control mass
should be zero.
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Conservation of Momentum

• Let B momentum of the system, mV.

The rate of change of momentum for a control mass
should be equal to resultant external force.
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First Law of Thermodynamics

• Let B E, Energy of the system, me.

The rate of change of energy of a control mass
should be equal to difference of work and heat
transfer rates.
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Rate Equations for Laws of Nature Control Mass
Conservation of Mass
Conservation of Momentum
First law of thermodynamics
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First Law for A Control Volume

• Conservation of mass

• Conservation of momentum

• Conservation of energy

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More Mathematical Definitions for A CV
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Thermodynamic Nature of Variables of CV

• Incoming and outgoing mass flow rates are steady.
• Properties of incoming and outgoing flows are
  homogeneous and invariant.
• Properties of CV can be inhomogeneous and
  variant.
• Following features for CV are possible.
• Inhomogeneous and variant Difficult to solve
  using thermodynamics.
• Homogeneous and invariant A trivial situation
  for a CV. No heat or work interactions required.
• Inhomogeneous and invariant Steady State System.
  
• Rate of work and heat interactions must be
  invariant too.
• Homogeneous and variant Transient System.
• Rate of work and heat interactions are variant.

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Applications of CV Analysis

• A means to estimate the size of engineering
  devices.

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First Law for CVSteady State Steady Flow
Properties of CV are Invariant

• Conservation of mass

NO accumulation or depletion of mass of a CV.

• Conservation of energy

NO addition or removal of energy for a CV.
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Rate of Work and Heat Transfers SSSF
Both rate of heat transfer and rate of work
transfer are invariant. The work done per unit
mass and heat transfer per unit mass are
invariant. The specific work transfer at
various parts of a CV can be different. The
specific heat transfer at various parts of a CV
can be different.
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The Steam Power Plant
Executes a Thermodynamic Cycle using an assembly
of CVs
Multiple Inflows Multiple Outlets !!!
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Complex Engineering Control Volume SSSF
SSSF Conservation of mass
First Law
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Comparison of A control mass and SSSF CV during a
change of state
Consider compression processes using CM and CV
devices.
Reciprocating compressor A Control Mass mCM
Initial State p1,v1 and T1.
Final State p2,v2 and T2.
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Centrifugal compressor A Control Volume _at_SSSF
Outlet State pout,vout and Tout.
Inlet State pin,vin and Tin.
Various parts of A CV are at different states
during SSSF process !!
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Salient Features of CV _at_SSSF Process

• Rate of mass inflow Rate mass outflow.
• Work done per unit mass is invariant.
• Heat transfer per unit mass is invariant.
• Change of state or process is not for the CV!
• The incoming fluid changes its state from inlet
  to exit conditions.
• A CM possesses different states at different time
  intervals.
• A CV possesses All states at any time but at
  different spatial locations.
• A CV with SSSF process is an inhomogeneous
  device.
• A CV can work continuously without changing its
  state.
• A CV lowers the importance of time !