Answer:
a) ΔEC=-23.4kW
b)W=12106.2kW
c)![A=0.01297m^2](https://tex.z-dn.net/?f=A%3D0.01297m%5E2)
Explanation:
A)
The kinetic energy is defined as:
(vel is the velocity, to differentiate with v, specific volume).
The kinetic energy change will be: Δ (
)=![\frac{m*vel_2^2}{2}-\frac{m*vel_1^2}{2}](https://tex.z-dn.net/?f=%5Cfrac%7Bm%2Avel_2%5E2%7D%7B2%7D-%5Cfrac%7Bm%2Avel_1%5E2%7D%7B2%7D)
Δ (
)=![\frac{m}{2}*(vel_2^2-vel_1^2)](https://tex.z-dn.net/?f=%5Cfrac%7Bm%7D%7B2%7D%2A%28vel_2%5E2-vel_1%5E2%29)
Where 1 and 2 subscripts mean initial and final state respectively.
Δ(
)=![\frac{12\frac{kg}{s}}{2}*(50^2-80^2)\frac{m^2}{s^2}=-23400W=-23.4kW](https://tex.z-dn.net/?f=%5Cfrac%7B12%5Cfrac%7Bkg%7D%7Bs%7D%7D%7B2%7D%2A%2850%5E2-80%5E2%29%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%3D-23400W%3D-23.4kW)
This amount is negative because the steam is losing that energy.
B)
Consider the energy balance, with a neglective height difference: The energy that enters to the turbine (which is in the steam) is the same that goes out (which is in the steam and in the work done).
![H_1+\frac{m*vel_1^2}{2}=H_2+\frac{m*vel_2^2}{2}+W\\W=m*(h_1-h_2)+\frac{m}{2} *(vel_1^2-vel_2^2)](https://tex.z-dn.net/?f=H_1%2B%5Cfrac%7Bm%2Avel_1%5E2%7D%7B2%7D%3DH_2%2B%5Cfrac%7Bm%2Avel_2%5E2%7D%7B2%7D%2BW%5C%5CW%3Dm%2A%28h_1-h_2%29%2B%5Cfrac%7Bm%7D%7B2%7D%20%2A%28vel_1%5E2-vel_2%5E2%29)
We already know the last quantity:
=
Δ (
)=23400W
For the steam enthalpies, review the steam tables (I attach the ones that I used); according to that, ![h_1=h(T=500C,P=4MPa)=3445.3\frac{kJ}{kg}](https://tex.z-dn.net/?f=h_1%3Dh%28T%3D500C%2CP%3D4MPa%29%3D3445.3%5Cfrac%7BkJ%7D%7Bkg%7D)
The exit state is a liquid-vapor mixture, so its enthalpy is:
![h_2=h_f+xh_{fg}=289.23+0.92*2366.1=2483.4\frac{kJ}{kg}](https://tex.z-dn.net/?f=h_2%3Dh_f%2Bxh_%7Bfg%7D%3D289.23%2B0.92%2A2366.1%3D2483.4%5Cfrac%7BkJ%7D%7Bkg%7D)
Finally, the work can be obtained:
![W=12\frac{kg}{s}*(3445.3-2438.4)\frac{kJ}{kg} +23.400kW)=12106.2kW](https://tex.z-dn.net/?f=W%3D12%5Cfrac%7Bkg%7D%7Bs%7D%2A%283445.3-2438.4%29%5Cfrac%7BkJ%7D%7Bkg%7D%20%2B23.400kW%29%3D12106.2kW)
C) For the area, consider the equation of mass flow:
where
is the density, and A the area. The density is the inverse of the specific volume, so ![m=\frac{vel*A}{v}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7Bvel%2AA%7D%7Bv%7D)
The specific volume of the inlet steam can be read also from the steam tables, and its value is:
, so:
![A=\frac{m*v}{vel}=\frac{12\frac{kg}{s}*0.08643\frac{m^3}{kg}}{80\frac{m}{s}}=0.01297m^2](https://tex.z-dn.net/?f=A%3D%5Cfrac%7Bm%2Av%7D%7Bvel%7D%3D%5Cfrac%7B12%5Cfrac%7Bkg%7D%7Bs%7D%2A0.08643%5Cfrac%7Bm%5E3%7D%7Bkg%7D%7D%7B80%5Cfrac%7Bm%7D%7Bs%7D%7D%3D0.01297m%5E2)