kill(all)$
load(lrats)$
load(vect)$

Basic idea of jet propulsion ,version 1.0
peter.vlasschaert@  gmail.com,27/09/2018

principle : jet propulsion ( also gas turbines).

1e) suppose : system P[O] >P[a]
===========================

cylinder with gas inside ( cross section = A)

P[O] = P[a] = pressure 

assume(P[O]>P[a]);

force : τ

p1:τ = (P[O]-P[a])*A;

2e) experiment : removes  ' gate 2 '
=============================

 open gate '2'

result  : pressure  P[0]  goes down : P (until τ=0)
               exit pressure (gate '2') result : velocity = u[e] = (e)xit velocity

p2:P[O]=P;
p2a:subst([p2],p1);

concept : limit process  :  P => P[a]
----------------------------------------------------

p2b:'limit(lhs(p2a),P,P[a]) = 'limit(rhs(p2a),P,P[a]);

p2c: [P=P,P[a]=Pa];
p2d:subst(p2c,p2b);

p2e:lhs(p2d)=ev(rhs(p2d),limit);

force : τ  => 0

3e)  to hold  P[i] = Thruster , Maintain the pressure.
==========================================

compensation pressure : P[i]

inlet pressure : loss of pressure = P[i] 

p3:τ = (p[i]-p[a])*A;

newton law :  F = m*a 
==================
=>  mass change  : τ = d M/dt =d(mv)/dt
m : mass
v  : velocity.
M : m*v

depends([v,m],t);

p4:M=m*v;
p4a:'diff(lhs(p4),t) =diff(rhs(p4),t);

suppose : no acceleration v => dv/dt =0
------------------------------------------------------------
=> result : 'constant : Thrust'.
                    ===============

p4b:[diff(v,t)=0];
p4c:subst(p4b,p4a);

definition : newton's law with mass change.
=====================================

p4d:lhs(p3) = rhs(p4c);

from Figure v = u[e]

p4e:[v=u[e]];

result : p4d

p4f:subst(p4e,p4d);

total result : 'p4f' = 'p3'

p5:rhs(p3)=rhs(p4f);

question : 'to find : u[e] '
====================

1e) cylinder with radius (= r) 
-----------------------------------------

p5a: [A= π*r^2];

2e) solve 'p5', u[e] ,use 1e)
----------------------------------------

p5b:subst(p5a,p5);

p5c:solve(p5b,u[e])[1];

rem : rate mass exit per time , is given  = dm/dt
u[e] = ' rocket ' = velocity nozzle exit.

question    ?                     :  given : u[e] , dm/dt , r ,P[a ]
                                             =======================
                                              =>   ' find p[i] '      ?
-------------------------------------------------------------------------------
rem : P[a] : given atmospheric pressure .
-------------------------------------------------------------------------------

  = >     solve ,use 'p5b'.  ?
======================

p6:solve(p5b,p[i])[1];

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