Arjan82 said:
This is a really disheartening thing to say, especially because in his very long post #100
@erobz gives in his equation 10 literally his answer to your question. It
literally gives you his equation of the mass flow rate, he's only asking you to do a bit of effort and plug in the numbers yourself... Don't you see that?
They are actually saying I didn't go far enough with that "huge working". I think it's their presentation of "you're wrong and I'm right, but I can't tell you why" is what strikes a nerve with everyone they seem to interact with...
That being said, In this case (after getting some sleep) It dawned on me that we have another equation that starts the whole process over again. This time it's the cliff notes version.
Assumption:
The gas is undergoing an adiabatic expansion from A to C ( ? I have less confidence that there is no heat transfer with the surroundings between A and C as I do in gong from A to B )
Hydrostatics:
$$ P_C = P_A + \rho_w g ( l_A - l_C ) $$
First Law:
$$ 0 = h_A - h_C + \frac{V^2_A- V^2_C}{2}$$
$$ \implies V_C = \sqrt{2 \left( h_A(T_A) - h_C(T_C) \right) + V^2_A} $$
Adiabatic Expansion:
$$ T_C = T_A \left( \frac{P_C}{P_A} \right)^{ \frac{1-\gamma}{\gamma}} = T_A \left( \frac{ P_A + \rho_w g ( l_A - l_C ) }{P_A} \right)^{ \frac{1-\gamma}{\gamma}} $$
Conservation of Mass (Steady State):
$$\dot m = \rho_C A V_C = \rho_A A V_A $$
$$ \implies \rho_C V_C = \rho_A V_A $$
Sub all that in along with the Ideal Gas Law:
$$ \frac{P_A + \rho_w g ( l_A - l_C )}{R T_A \left( \frac{ P_A + \rho_w g ( l_A - l_C ) }{P_A} \right)^{ \frac{1-\gamma}{\gamma}} } \sqrt{2 \left( h_A(T_A) - h_C(T_C) \right) + V^2_A} = \frac{P_A}{RT_A} V_A $$
Where:
##T_C = T_A \left( \frac{P_C}{P_A} \right)^{ \frac{1-\gamma}{\gamma}} = T_A \left( \frac{ P_A + \rho_w g ( l_A - l_C ) }{P_A} \right)^{ \frac{1-\gamma}{\gamma}}##
##T_B = T_A \left( \frac{P_B}{P_A} \right)^{ \frac{1-\gamma}{\gamma}} = T_A \left( \frac{ P_A + \rho_w g ( l_A - l_B ) }{P_A} \right)^{ \frac{1-\gamma}{\gamma}}##
##V_A = \sqrt{ \frac{ 2 \left( h_B(T_B) - h_A(T_A) \right) }{ 1 - \left( \frac{P_A}{P_B} \right)^{ \frac{2}{\gamma}} \left( \frac{A}{a} \right)^2 } }##
## P_B = P_A + \rho_w g ( l_A - l_B) ##
## h_B(T_B) = \int_{0}^{T_B} c_p(T)dT ##
## h_C(T_C) = \int_{0}^{T_C} c_p(T)dT ##
"In theory" "you might be able to determine ##P_A##, and then ##\dot m##, but ##P_A## is not only buried in a sea of symbols, but also some of those symbols ##h_B(T_B), h_C(T_C)## it finds itself buried in a non-linear function, which is then buried in a limit of an integral...It's a real pickle of an equation.
At this point I fully expect
@Sailor Al to say , "but I still don't see an answer to my question", To which I say
