Answer:
a
The rate of work developed is 
b
The rate of entropy produced within the turbine is
Explanation:
From the question we are told
The rate at which heat is transferred is 
the negative sign because the heat is transferred from the turbine
The specific heat capacity of air is 
The inlet temperature is 
The outlet temperature is 
The pressure at the inlet of the turbine is 
The pressure at the exist of the turbine is 
The temperature at outer surface is 
The individual gas constant of air R with a constant value 
The general equation for the turbine operating at steady state is \

h is the enthalpy of the turbine and it is mathematically represented as

The above equation becomes


Where
is the heat transfer from the turbine
is the work output from the turbine
is the mass flow rate of air
is the rate of work developed
Substituting values


The general balance equation for an entropy rate is represented mathematically as

=> 
generally ![(s_1 -s_2) = \Delta s = c_p\ ln[\frac{T_2}{T_1} ] + R \ ln[\frac{v_2}{v_1} ]](https://tex.z-dn.net/?f=%28s_1%20-s_2%29%20%3D%20%5CDelta%20s%20%3D%20c_p%5C%20ln%5B%5Cfrac%7BT_2%7D%7BT_1%7D%20%5D%20%2B%20R%20%5C%20ln%5B%5Cfrac%7Bv_2%7D%7Bv_1%7D%20%5D)
substituting for
![\frac{\sigma}{\r m} = \frac{-\r Q}{\r m} * \frac{1}{T_s} + c_p\ ln[\frac{T_2}{T_1} ] - R \ ln[\frac{p_2}{p_1} ]](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csigma%7D%7B%5Cr%20m%7D%20%3D%20%5Cfrac%7B-%5Cr%20Q%7D%7B%5Cr%20m%7D%20%2A%20%5Cfrac%7B1%7D%7BT_s%7D%20%2B%20%20c_p%5C%20ln%5B%5Cfrac%7BT_2%7D%7BT_1%7D%20%5D%20-%20R%20%5C%20ln%5B%5Cfrac%7Bp_2%7D%7Bp_1%7D%20%5D)
Where
is the rate of entropy produced within the turbine
substituting values
![\frac{\sigma}{\r m} = - (-30) * \frac{1}{315} + 1.1 * ln\frac{670}{970} - 0.287 * ln [\frac{100kPa}{400kPa} ]](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csigma%7D%7B%5Cr%20m%7D%20%3D%20-%20%28-30%29%20%2A%20%5Cfrac%7B1%7D%7B315%7D%20%2B%201.1%20%2A%20ln%5Cfrac%7B670%7D%7B970%7D%20-%200.287%20%2A%20ln%20%5B%5Cfrac%7B100kPa%7D%7B400kPa%7D%20%5D)