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Quasi - One Dimensional Flow with Heat Addition

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Quasi - One Dimensional Flow with Heat Addition P M V Subbarao Professor Mechanical Engineering Department I I T Delhi A Gas Dynamic Model for Combustion Systems .. – PowerPoint PPT presentation

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Title: Quasi - One Dimensional Flow with Heat Addition


1
Quasi - One Dimensional Flow with Heat Addition
  • P M V Subbarao
  • Professor
  • Mechanical Engineering Department
  • I I T Delhi

A Gas Dynamic Model for Combustion Systems ..
2
Variable Area with Heat Transfer
Conservation of mass for steady flow
Conservation of momentum for ideal steady flow
3
Conservation of energy for ideal steady flow
Ideal Gas law
Combining momentum and gas law
4
Using conservation of mass
5
Mach number equation
6
Energy Equation with Mach Equation
7
Combined momentum,mass, gas Mach Equations
8
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9
Condition for M1
10
For heat addition, M1, dA will be positive. For
heat removal, M1, dA will be negative.
11
Constant Mach Number Flow with Heat Transfer
12
Quasi - One Dimensional Flow with Heat Transfer
Friction
  • P M V Subbarao
  • Professor
  • Mechanical Engineering Department
  • I I T Delhi

A Gas Dynamic Model for Gas Cooled High Heat
Release Systems..
13
Frictional Flow with Heat Transfer
14
Governing Equations
Nonreacting, no bodyforces, viscous work
negligible
Conservation of mass for steady flow
Conservation of momentum for frictional steady
flow
Conservation of energy for ideal steady flow
15
Ideal Gas law
Mach number equation
16
Into momentum equation
17
Combine conservation, state equations can
algebraically show
So we have three ways to change M of flow area
change (dA) previously studied friction f gt
0, same effect as dA heat transferheating,
q gt 0, like dA cooling, q
lt 0, like dA
18
Mach Number Variations
  • Subsonic flow (Mlt1) 1M2 gt 0
  • friction, heating, converging area increase M
    (dM gt 0)
  • cooling, diverging area decrease M (dM lt 0)
  • Supersonic flow (Mgt1) 1M2 lt 0
  • friction, heating, converging area decrease M
    (dM lt 0)
  • cooling, diverging area increase M (dM gt 0)

19
Sonic Flow Trends
Friction accelerates subsonic flow,
decelerates supersonic flow always drives flow
toward M1 (increases entropy) Heating same
as friction - always drives flow toward M1
(increases entropy) Cooling opposite - always
drives flow away from M1 (decreases entropy)
20
Nozzles Sonic Throat
Effect on transition point sub ? supersonic
flow As M?1, 1M2?0, need term to approach
0 For isentropic flow, previously showed
sonic condition was dA0, throat For friction
or heating, need dA gt 0 sonic point in
diverging section For cooling, need dA lt 0
sonic point in converging section
21
Mach Number Relations
Using conservation/state equations can get
equations for each TD property as function of dM2
22
Constant Area, Steady Compressible Flow
withFriction Factor and Uniform Heat Flux at the
Wall Specified
  • Choking limits and flow variables for passages
    are important parameters in one-dimensional,
    compressible flow in heated pipes.
  • The design of gas cooled beam stops and gas
    cooled reactor cores, both usually having helium
    as the coolant and graphite as the heated wall.
  • Choking lengths are considerably shortened by
    wall heating.
  • Both the solutions for adiabatic and isothermal
    flows overpredict these limits.
  • Consequently, an unchoked cooling channel
    configuration designed on the basis of adiabatic
    flow maybe choked when wall heat transfer is
    considered.

23
Gas Cooled Reactor Core
24
Beam Coolers
25
  • The local Mach number within the passage will
    increase towards the exit for either of two
    reasons or a combination of the two.
  • Both reasons are the result of a decrease in gas
    density with increasing axial position caused
    either by
  • (1) a frictional pressure drop or
  • (2) an increase in static temperature as a
    result of wall heat transfer.

Constant area duct
26
Divide throughout by dx
27
Multiply throughout by M2
For a uniform wall heat flux q
28
Numerical Integration of differential Equation
29
Choking Length
K non dimensional heat flux
M1
30
Mach number equation
31
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32
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