The Absorption of Heat by Solids and Liquids — Question 6

Question 6

Condensing steam supplies two distinct energy contributions

Steam at $100^\circ$C is injected into .400 kg water at $20.0^\circ$C in an insulated cup of negligible heat capacity. All injected steam condenses, and the final temperature is $60.0^\circ$C. Use $L_v=2.26\times10^6\,\mathrm{J/kg}$ and $c_w=4200\,\mathrm{J/(kg\,K)}$. Ignore steam kinetic energy and losses.

1. Find the required injected steam mass, including cooling of the condensate.

2. Find the result if condensate cooling is omitted and the resulting percentage overestimate of steam mass.

Question 6 - Solution

Condensing steam supplies two distinct energy contributions

(a) Method and calculation

$$m_s[L_v+c_w(100-60)]=.400c_w(60-20).$$$$m_s=\frac{67200}{2260000+168000}=.0276771\,\mathrm{kg}.$$Each kilogram releases latent heat and then sensible heat as the condensate cools.

(b) Interpretation and further calculation

$$m_{s,\mathrm{latent\ only}}=67200/2260000=.0297345\,\mathrm{kg}.$$$$\frac{m_{s,\mathrm{latent\ only}}-m_s}{m_s}\times100\%=\frac{168000}{2260000}\times100\%\approx7.4336\%.$$Omitting a real energy contribution overestimates the required source mass.

Physical check

The final temperature is below the condensation temperature, consistent with fully liquid retained water.