Answer:
the lost work per kilogram of water for this everyday household happening = 0.413 kJ/kg
Explanation:
Given that:
Initial Temperature
= 15°C
Initial Pressure
= 5 atm
Final Pressure
= 1 atm
Data obtain from steam tables of saturated water at 15°C are as follows:
Specific volume v = 1.001 cm³/gm
The change in temperature = 2°C
Specific heat of water = 4.19 J/gm.K
volume expansivity β = 1.5 × 10⁻⁴ K⁻¹
The expression to determine the change in temperature can be given as :
![\delta \ T = \frac{-V (1- \beta \ T}{C_p} * \delta \ P ( \frac{1}{9.87} \ \frac{J}{cm^3/atm})](https://tex.z-dn.net/?f=%5Cdelta%20%5C%20T%20%3D%20%5Cfrac%7B-V%20%281-%20%5Cbeta%20%5C%20T%7D%7BC_p%7D%20%2A%20%5Cdelta%20%5C%20P%20%28%20%5Cfrac%7B1%7D%7B9.87%7D%20%5C%20%5Cfrac%7BJ%7D%7Bcm%5E3%2Fatm%7D%29)
![\delta \ T = \frac{-1.001 \frac{cm^3}{gm} (1- 1.5*10^{-4} \ K^{-1} )*2}{4.19 \ \frac{J}{gm.K}} *(5-1)atm ( \frac{1}{9.87} \ \frac{J}{cm^3/atm})](https://tex.z-dn.net/?f=%5Cdelta%20%5C%20T%20%3D%20%5Cfrac%7B-1.001%20%5Cfrac%7Bcm%5E3%7D%7Bgm%7D%20%281-%201.5%2A10%5E%7B-4%7D%20%5C%20%20K%5E%7B-1%7D%20%29%2A2%7D%7B4.19%20%5C%20%5Cfrac%7BJ%7D%7Bgm.K%7D%7D%20%2A%285-1%29atm%20%28%20%5Cfrac%7B1%7D%7B9.87%7D%20%5C%20%5Cfrac%7BJ%7D%7Bcm%5E3%2Fatm%7D%29)
Δ T = 0.093 K
Now; we can calculate the lost work bt the formula:
![W_{lost} = T_{surr} *S](https://tex.z-dn.net/?f=W_%7Blost%7D%20%3D%20T_%7Bsurr%7D%20%2AS)
where ;
is the temperature of the surrounding. = 20°C = (20+273.15)K = 293.15 K
From above the change in entropy is:
![\delta \ S = C_p \ In (\frac{T+ \delta \ T }{T}) * \beta V \delta P](https://tex.z-dn.net/?f=%5Cdelta%20%5C%20%20S%20%3D%20C_p%20%5C%20%20In%20%28%5Cfrac%7BT%2B%20%5Cdelta%20%5C%20T%20%7D%7BT%7D%29%20%2A%20%20%5Cbeta%20V%20%5Cdelta%20P)
![\delta \ S = 4.19* \ In (\frac{288.15+0.093 }{288.15}) - 1.5*10^{-4} * 1.001 (5-1)* (\frac{1}{9.87})](https://tex.z-dn.net/?f=%5Cdelta%20%5C%20%20S%20%3D%204.19%2A%20%20%5C%20%20In%20%28%5Cfrac%7B288.15%2B0.093%20%7D%7B288.15%7D%29%20-%20%201.5%2A10%5E%7B-4%7D%20%2A%201.001%20%285-1%29%2A%20%28%5Cfrac%7B1%7D%7B9.87%7D%29)
![\delta \ S =1.408*10^{-3} \ J/gm.K](https://tex.z-dn.net/?f=%5Cdelta%20%5C%20%20S%20%3D1.408%2A10%5E%7B-3%7D%20%5C%20J%2Fgm.K)
![W_{lost} = T_{surr} *S](https://tex.z-dn.net/?f=W_%7Blost%7D%20%3D%20T_%7Bsurr%7D%20%2AS)
![W_{lost} = 293.15* 1.408*10^{-3} \ J/gm.K](https://tex.z-dn.net/?f=W_%7Blost%7D%20%3D%20293.15%2A%201.408%2A10%5E%7B-3%7D%20%5C%20J%2Fgm.K)
![W_{lost} = 0.413 \ kJ/kg](https://tex.z-dn.net/?f=W_%7Blost%7D%20%3D%200.413%20%5C%20%20kJ%2Fkg)
Thus, the lost work per kilogram of water for this everyday household happening = 0.413 kJ/kg