<u>Answer:</u> The rate of heat flow is 3.038 kW and the rate of lost work is 1.038 kW.
<u>Explanation:</u>
We are given:
![C_p=\frac{7}{2}R\\\\T=475K\\P_1=100kPa\\P_2=50kPa](https://tex.z-dn.net/?f=C_p%3D%5Cfrac%7B7%7D%7B2%7DR%5C%5C%5C%5CT%3D475K%5C%5CP_1%3D100kPa%5C%5CP_2%3D50kPa)
Rate of flow of ideal gas , n = 4 kmol/hr =
(Conversion factors used: 1 kmol = 1000 mol; 1 hr = 3600 s)
Power produced = 2000 W = 2 kW (Conversion factor: 1 kW = 1000 W)
We know that:
(For isothermal process)
So, by applying first law of thermodynamics:
![\Delta U=\Delta q-\Delta W](https://tex.z-dn.net/?f=%5CDelta%20U%3D%5CDelta%20q-%5CDelta%20W)
.......(1)
Now, calculating the work done for isothermal process, we use the equation:
![\Delta W=nRT\ln (\frac{P_1}{P_2})](https://tex.z-dn.net/?f=%5CDelta%20W%3DnRT%5Cln%20%28%5Cfrac%7BP_1%7D%7BP_2%7D%29)
where,
= change in work done
n = number of moles = 1.11 mol/s
R = Gas constant = 8.314 J/mol.K
T = temperature = 475 K
= initial pressure = 100 kPa
= final pressure = 50 kPa
Putting values in above equation, we get:
![\Delta W=1.11mol/s\times 8.314J\times 475K\times \ln (\frac{100}{50})\\\\\Delta W=3038.45J/s=3.038kJ/s=3.038kW](https://tex.z-dn.net/?f=%5CDelta%20W%3D1.11mol%2Fs%5Ctimes%208.314J%5Ctimes%20475K%5Ctimes%20%5Cln%20%28%5Cfrac%7B100%7D%7B50%7D%29%5C%5C%5C%5C%5CDelta%20W%3D3038.45J%2Fs%3D3.038kJ%2Fs%3D3.038kW)
Calculating the heat flow, we use equation 1, we get:
[ex]\Delta q=3.038kW[/tex]
Now, calculating the rate of lost work, we use the equation:
![\text{Rate of lost work}=\Delta W-\text{Power produced}\\\\\text{Rate of lost work}=(3.038-2)kW\\\text{Rate of lost work}=1.038kW](https://tex.z-dn.net/?f=%5Ctext%7BRate%20of%20lost%20work%7D%3D%5CDelta%20W-%5Ctext%7BPower%20produced%7D%5C%5C%5C%5C%5Ctext%7BRate%20of%20lost%20work%7D%3D%283.038-2%29kW%5C%5C%5Ctext%7BRate%20of%20lost%20work%7D%3D1.038kW)
Hence, the rate of heat flow is 3.038 kW and the rate of lost work is 1.038 kW.