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 …..

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

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

Using conservation of mass

Mach number equation:

Energy Equation with Mach Equation:

Combined momentum,mass, gas & Mach Equations

Condition for M=1

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

Constant Mach Number Flow with Heat Transfer

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…..

Frictional Flow with Heat Transfer

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:

Ideal Gas law: Mach number equation:

Into momentum equation

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

Mach Number Variations Subsonic flow (M 0 – friction, heating, converging area increase M (dM > 0) – cooling, diverging area decrease M (dM < 0) Supersonic flow (M>1): 1–M 2 < 0 – friction, heating, converging area decrease M (dM < 0) – cooling, diverging area increase M (dM > 0)

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 – (increases entropy) Cooling – opposite - always drives flow away from M=1 – (decreases entropy)

Nozzles : Sonic Throat Effect on transition point: sub  supersonic flow As M  1, 1–M 2  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

Mach Number Relations Using conservation/state equations can get equations for each TD property as function of dM 2

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 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.

Gas Cooled Reactor Core

Beam Coolers

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:

Divide throughout by dx

Multiply throughout by M 2 For a uniform wall heat flux q’’

Numerical Integration of differential Equation

Choking Length M1M1 K :non dimensional heat flux

Mach number equation: