1. One Dimensional Flow with Heat Addition
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
A Gas Dynamic Model for Cross Country Gas Pipe Lines…..

2. Ideal Flow in A Constant Area Duct with heat Transfer

3. Mach equation gives
Mach equation in Momentum equation gives
Mach equation in Energy equation gives

4. Change over A Finite Length2 1

5. Integrating from point 1 to point 2:
In subsonic flow, heat addition increases the Mach Number.
In supersonic flow, heat addition decreases the Mach Number.
Addition of heat leads the flow to move towards M=1.
Removal of heat leads the flow to move away from M=1.
Therefore T0 will be maximum when M=1.

7. Variation of Stagnation Temperature with Mach Number
Heat Removal
Heat Removal T0
Heat Addition
Heat Addition
M=1 M

8. Total Heat Addition or Removal
Total heat transfer per unit mass flow rate

10. Relation Between M1 and M2
Adiabatic ideal flow:

11. Similarly, for an ideal flow with heat addition
and for an ideal flow with heat removal

12. Relation Between M1 and M2
Heat Addition
Adiabatic
Heat Removal M2 M1

13. Entropy Change : Heat Addition or Removal
Integration from 1 to 2:

15. Adiabatic
Heat Removal
Heat Addition M1

16. Maximum End Condition
If heat is added to the flow, the Mach number tends towards one.
If heat is removed from the flow, the Mach number tends away from one.
All the properties of the flow can be conveniently written in terms of conditions that exist when M2 = 1.

18. Variation of Normalized Properties
Normalized Values M1

19. Temperature Entropy Relation
Traditionally, heat addition or removal is characterized through relative temperature – entropy variations.
Entropy signifies the quality of heat transfer process.
An explicit relation between entropy and temperature is very useful in evaluating the heat transfer process.
On integration till maximum end point.

20. The pressure ratio equation gives:
Substitute M2 in equation for temperature ratio :

21. The roots of the equation are:
This allows the variation of temperature ratio with change in entropy to be found for any value of g.

22. Rayliegh Line
One dimensional ideal flow with heat transfer is called as Rayliegh flow.

23. Maximum Entropy and Maximum Temperature Points
Entropy will be maximum when M=1.
Heat addition moves the Mach number towards 1 and vice versa.
The point of maximum temperature occurs not at M=1.
This value can be found by differentiating temperature ratio equation.

24. M corresponding to Tmax:
Tmax occurs at M<1 and

25. Summary of the Effect of Heat Transferq T0
M & V p s
M<1 + -
M>1

26. Variable Area with Heat Transfer
Conservation of mass for steady flow:
Conservation of momentum for ideal steady flow:

27. Conservation of energy for ideal steady flow:
Ideal Gas law:
Combining momentum and gas law:

28. Using conservation of mass

29. Mach number equation:

30. Energy Equation with Mach Equation:

31. Combined momentum,mass, gas & Mach Equations

33. Condition for M=1

34. For heat addition, M=1, dA will be positive.
For heat removal, M=1, dA will be negative.

35. Constant Mach Number Flow with Heat Transfer

36. One Dimensional Flow with Heat Transfer & Friction
P M V Subbarao
Associate Professor
Mechanical Engineering Department
I I T Delhi
A Gas Dynamic Model for Gas Cooled High Heat Release Systems…..

37. Frictional Flow with Heat Transfer

38. 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:

39. Ideal Gas law:
Mach number equation:

40. Into momentum equation

41. Combine conservation, state equations– can algebraically show
So we have three ways to change M of flow
– area change (dA): previously studied
– friction: f > 0, same effect as –dA
– heat transfer:heating, q’’’ > 0, like –dA cooling, q’’’ < 0, like +dA

42. Mach Number Variations
Subsonic flow (M<1): 1–M2 > 0
– friction, heating, converging area increase M (dM > 0)
– cooling, diverging area decrease M (dM < 0)
• Supersonic flow (M>1): 1–M2 < 0
– friction, heating, converging area decrease M (dM < 0)
– cooling, diverging area increase M (dM > 0)

43. Sonic Flow Trends • Friction
– accelerates subsonic flow, decelerates supersonic flow
– always drives flow toward M=1
– (increases entropy)
• Heating
– same as friction - always drives flow toward M=1
• Cooling
– opposite - always drives flow away from M=1
– (decreases entropy)

44. • Effect on transition point: sub  supersonic flow
Nozzles : Sonic Throat
• Effect on transition point: sub  supersonic flow
• As M1, 1–M20, need { } term to approach 0
• For isentropic flow, previously showed
– sonic condition was dA=0, throat
• For friction or heating, need dA > 0
– sonic point in diverging section
• For cooling, need dA < 0
– sonic point in converging section

45. Mach Number Relations
• Using conservation/state equations can get equations for each TD property as function of dM2

46. Constant Area, Steady Compressible Flow with Friction 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
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.

47. Gas Cooled Reactor Core

48. Beam Coolers

49. 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:

50. Divide throughout by dx

51. Multiply throughout by M2
For a uniform wall heat flux q’’

52. Numerical Integration of differential Equation

53. Choking Length
K :non dimensional heat flux M1

54. Mach number equation: