Answer:
The rate at which entropy is produced within the turbine is 22.762 kilojoules per kilogram-Kelvin.
Explanation:
By either the Principle of Mass Conservation and First and Second Laws of Thermodynamics, we model the steam turbine by the two equations described below:
Principle of Mass Conservation
(1)
First Law of Thermodynamics
(2)
Second Law of Thermodynamics
(3)
By dividing each each expression by
, we have the following system of equations:
(2b)
(3b)
Where:
- Heat transfer rate between the turbine and its surroundings, in kilowatts.
- Specific heat transfer between the turbine and its surroundings, in kilojoules per kilogram.
- Outer surface temperature of the turbine, in Kelvin.
- Mass flow rate through the turbine, in kilograms per second.
,
- Specific enthalpy of water at inlet and outlet, in kilojoules per kilogram.
,
- Speed of water at inlet and outlet, in meters per second.
- Specific work of the turbine, in kilojoules per kilogram.
,
- Specific entropy of water at inlet and outlet, in kilojoules per kilogram-Kelvin.
- Specific generated entropy, in kilojoules per kilogram-Kelvin.
By property charts for steam, we get the following information:
Inlet
,
,
, ![s = 6.9235\,\frac{kJ}{kg\cdot K}](https://tex.z-dn.net/?f=s%20%3D%206.9235%5C%2C%5Cfrac%7BkJ%7D%7Bkg%5Ccdot%20K%7D)
Outlet
,
,
, ![s = 7.3542\,\frac{kJ}{kg\cdot K}](https://tex.z-dn.net/?f=s%20%3D%207.3542%5C%2C%5Cfrac%7BkJ%7D%7Bkg%5Ccdot%20K%7D)
If we know that
,
,
,
,
,
,
and
, then the rate at which entropy is produced withing the turbine is:
![q_{out} = h_{in} - h_{out} + \frac{1}{2}\cdot (v_{in}^{2}-v_{out}^{2})-w_{out}](https://tex.z-dn.net/?f=q_%7Bout%7D%20%3D%20h_%7Bin%7D%20-%20h_%7Bout%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20%28v_%7Bin%7D%5E%7B2%7D-v_%7Bout%7D%5E%7B2%7D%29-w_%7Bout%7D)
![q_{out} = 3231.7\,\frac{kJ}{kg} - 2675.6\,\frac{kJ}{kg} + \frac{1}{2}\cdot \left[\left(160\,\frac{m}{s} \right)^{2}-\left(100\,\frac{m}{s} \right)^{2}\right] - 540\,\frac{kJ}{kg}](https://tex.z-dn.net/?f=q_%7Bout%7D%20%3D%203231.7%5C%2C%5Cfrac%7BkJ%7D%7Bkg%7D%20-%202675.6%5C%2C%5Cfrac%7BkJ%7D%7Bkg%7D%20%2B%20%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20%5Cleft%5B%5Cleft%28160%5C%2C%5Cfrac%7Bm%7D%7Bs%7D%20%5Cright%29%5E%7B2%7D-%5Cleft%28100%5C%2C%5Cfrac%7Bm%7D%7Bs%7D%20%5Cright%29%5E%7B2%7D%5Cright%5D%20-%20540%5C%2C%5Cfrac%7BkJ%7D%7Bkg%7D)
![q_{out} = 7816.1\,\frac{kJ}{kg}](https://tex.z-dn.net/?f=q_%7Bout%7D%20%3D%207816.1%5C%2C%5Cfrac%7BkJ%7D%7Bkg%7D)
![s_{gen} = \frac{q_{out}}{T_{out}}+s_{out}-s_{in}](https://tex.z-dn.net/?f=s_%7Bgen%7D%20%3D%20%5Cfrac%7Bq_%7Bout%7D%7D%7BT_%7Bout%7D%7D%2Bs_%7Bout%7D-s_%7Bin%7D)
![s_{gen} = \frac{7816.1\,\frac{kJ}{kg} }{350\,K} + 7.3542\,\frac{kJ}{kg\cdot K} - 6.9235\,\frac{kJ}{kg\cdot K}](https://tex.z-dn.net/?f=s_%7Bgen%7D%20%3D%20%5Cfrac%7B7816.1%5C%2C%5Cfrac%7BkJ%7D%7Bkg%7D%20%7D%7B350%5C%2CK%7D%20%2B%207.3542%5C%2C%5Cfrac%7BkJ%7D%7Bkg%5Ccdot%20K%7D%20-%206.9235%5C%2C%5Cfrac%7BkJ%7D%7Bkg%5Ccdot%20K%7D)
![s_{gen} = 22.762\,\frac{kJ}{kg\cdot K}](https://tex.z-dn.net/?f=s_%7Bgen%7D%20%3D%2022.762%5C%2C%5Cfrac%7BkJ%7D%7Bkg%5Ccdot%20K%7D)
The rate at which entropy is produced within the turbine is 22.762 kilojoules per kilogram-Kelvin.