Question about Jet engines (thermodynamics)

[Maximilian Popelier]

Hi, I need help
I got a project for school about jet engines. They say bysubstitution of equations 5.1, 5.2, 5.5 and 5.6
butI don't know the steps how i get t0 the last formule.

Thank you
Maximilian Popelier

 

[DrClaude]

Take eq. (5.7) and replace $w_a$ and $q_a$ by the right-hand side of (5.5) and (5.6), and then further simplify using (5.1) and (5.2).

 

[Maximilian Popelier]

Tank you very much i will try it tomorrow

 

[Maximilian Popelier]

hi thank you..
but now I'm stuck here...

 

[DrClaude]

You made a mistake in the numerator. After having corrected that mistake, before you use equations (5.1) and (5.2), notice that η is 1 - something, so try to get that 1.

 

[Maximilian Popelier]

Im sorry I tried it but I am again stuck at the next step...

 

[DrClaude]

Im sorry I tried it but I am again stuck at the next step...

Please show your work.

 

[Maximilian Popelier]

My whole Project or just the formule?
Here is the formule:

 

[DrClaude]

You made a mistake in the last equality
$$
1 + \frac{-a}{b} = 1 - \frac{a}{b}
$$
$$
1 + \frac{-a}{b} \neq 1 - \frac{a}{-b}
$$

 

[Maximilian Popelier]

owww... sorry but I am still stuck

 

[DrClaude]

You can now take out a factor in front of the fraction.

 

[Maximilian Popelier]

How if i do it i have do divise the T1 by the Factor

 

[Maximilian Popelier]

i forgot the 1 -

 

[DrClaude]

Take a factor out of the numerator only.

 

[Maximilian Popelier]

oke and then?

 

[DrClaude]

oke and then?

Then you should get eq. (5.8). You can write what you got in post #13 such that you get the same term in $T_1$ and $T_3$ in the numerator and the denominator.

 

[Maximilian Popelier]

sorry i don't get it but i got this

 

[DrClaude]

sorry i don't get it but i got thisView attachment 109348

Compare what is in the parenthesis with the denominator.

I now see you also made a mistake when changing $T_4$ to $T_3$ in the numerator.

 

[Maximilian Popelier]

Nice i got it!
Ty very much

 

[Maximilian Popelier]

Oh i didn't get it i got a "-" wrong i will take a picture

 

[Maximilian Popelier]

 

[DrClaude]

$$T_4 = T_3 \left( \frac{p_2}{p_1} \right)^{-\left(1 - \frac{1}{k} \right)}$$ so you should get the right factor in front.

 

[Maximilian Popelier]

Can you please write it all out and take a picture. It will save us a lot of time

 

[DrClaude]

$$
\begin{align*}
\eta_{\mathrm{th}} &= 1 - \frac{T_4 - T_1}{T_3 - T_2} \\
&= 1 - \frac{T_3 \left( \frac{p_2}{p_1} \right)^{-\left(1 - \frac{1}{k} \right)} - T_1}{T_3 - T_1 \left( \frac{p_2}{p_1} \right)^{\left(1 - \frac{1}{k} \right)}} \\
&= 1 - \left( \frac{p_2}{p_1} \right)^{-\left(1 - \frac{1}{k} \right)} \frac{T_3 - T_1 \left( \frac{p_2}{p_1} \right)^{\left(1 - \frac{1}{k} \right)}}{T_3 - T_1 \left( \frac{p_2}{p_1} \right)^{\left(1 - \frac{1}{k} \right)}}
\end{align*}
$$

 

[Maximilian Popelier]

Ty man

 

[Maximilian Popelier]

Oh but you made a mistake i guess
Im not 100% sure

 

[DrClaude]

Oh but you made a mistake i guess

Why? I get the same equation as in the book.

 

[Maximilian Popelier]

 

[Maximilian Popelier]

Got it ty

 

[Maximilian Popelier]

Hi it isn't very clear for me sorry you hen you replace t4 u use a extra "-" before (1-1/k)
Sorry that i don't get it

 

[DrClaude]

Equation 5.4 is
$$
T_4 = T_3 / (p_2/p_1)^{(k-1)/k}
$$
so
$$

T_4 = T_3 \left( \frac{p_2}{p_1} \right)^{-(k-1)/k} = T_3 \left( \frac{p_2}{p_1} \right)^{-(1-\frac{1}{k})}
$$

 

[Maximilian Popelier]

Ty dude