# Homework check for On/Off Design of a Fixed Area Turbojet

## Homework Statement

An ideal fixed-area turbojet is operated where $$\pi_c=15,M_o=0.8,T_o=260K, T_{t4}=2000 K$$, and $$P_o=20000 Pa$$. Mass flow rate of air processed by this engine at on-design is 100 kg/sec.

What will be the performance of this engine (thrust, fuel consumption) compared to the on-design conditions if it is flown at a Mach of 0.3 and at an altitude where temperature and pressure are 288K and 101325 Pa. Furthermore, the fuel throttle is set such that $$T_{t4}=1500 K$$ at this off-design point. Assume that $$A_9$$ is varied to keep $$P_9=P_o$$.

## Homework Equations

$$\tau_r=1+\frac{(\gamma-1)}{2}M_o^2$$

$$\pi_r=\tau_r^{\frac{\gamma}{\gamma-1}}$$

$$\tau_c=\pi_c^{\frac{\gamma-1}{\gamma}}$$

$$\tau_\lambda=\frac{T_{t4}}{T_o}$$

$$\pi_t=\tau_t^{\frac{\gamma}{\gamma-1}}$$

$$\frac{T_{t3}}{T_o}=\tau_r{\tau_c}$$

$$a_o=\sqrt{\gamma{R}{T_o}}$$

$$U_o=a_o{M_o}$$

$$U_9=a_o\left[\frac{2}{\gamma-1}{\tau_\lambda}{\tau_t}\left[1-\left({\pi_r}{\pi_d}{\pi_c}{\pi_b}{\pi_t}{\pi_N}\right)^{-\frac{(\gamma-1)}{\gamma}}\right]\right]^{1/2}$$

$$\frac{THRUST}{\dot{m}}=U_9-U_o$$

$$\dot{m_f}h=(\dot{m_a}+\dot{m_f})C_p{T_{t4}}-\dot{m_a}C_o{T_{t3}}$$

## The Attempt at a Solution

So the givens for on-design analysis:

$$\pi_c=15$$

$$M_o=0.8$$

$$T_o=260 K$$

$$T_{t4}=2000 K$$

$$P_o=20,000 Pa$$

$$\dot{m_a}=100 kg/s$$

$$h=4.5 *10^7$$

$$C_p=1004$$

Solving for each variable I get the following:

$$\tau_r=1.128$$

$$\pi_r=1.524$$

$$\tau_c=2.168$$

$$\tau_\lambda=7.69$$

$$\tau_t=.829$$

$$\pi_t=.519$$

Re-arranging $$\frac{T_{t3}}{T_o}$$ to solve for $$T_{t3}$$ I get 635.83 K

$$a_o=322.65 m/s$$

$$U_o=258.12 m/s$$

Assuming unknown $$\pi{'s}=1$$, then

$$U_9=1296.38 m/s$$

Solving for THRUST I get 103,826 N

And finally

$$\dot{m_f}=3.185 kg/s$$

Now for the off-design:

$$M_o=0.3$$

$$T_o=288 K$$

$$T_{t4}=1500 K$$

$$P_o=101325 Pa$$

$$h=4.5 *10^7$$

$$C_p=1004$$

For off-design we keep $$\tau_t$$ the same value for on-design, so
$$\tau_t=.829$$

$$\pi_t=.519$$

Solving for each variable I get the following:

$$\tau_r=1.018$$

$$\pi_r=1.064$$

$$/tau_c=1.875$$

$$\tau_\lambda=5.21$$

$$T_{t3}=549.72 K$$

$$a_o=339.58 m/s$$

$$U_o=101.87 m/s$$

$$U_9=957.36 m/s$$

And now I need to calculate thrust but I need to find $$\dot{m_a}$$ first.

Also I found what the area is using $$A_o=\frac{\dot{m_a}}{\rho_o{R}{U_o}}$$ which came out to be .00504 m^2. This seems like a small area for the turbojet. Did I mess something up?

If I use this area value to find $$\dot{m_f}$$ for the off-design analysis using $$\rho=\frac{P_o}{RT_o}=1.23$$ at this new pressure and temperature I get the mass flow rate of air to be 457.7 kg/s. Does this value make since? To me it doesn't because of the slower velocity in the off-design vs. the higher velocity in the on-design Then again the density for the on-design case is much lower than the off-design (.268 kg/m^3 for on-design).

The mass flow rate of fuel is then 10.04 kg/s. I would be grateful if someone could look over this and see if I made an error in logic or calculations.

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chemisttree