Question

For offices to feel comfortable, we like the relative humidity indoors to be about 50% and the temperature to be about 20◦C. An office is normally ventilated: out- side air flows through the office, where it is heated by a radiator. At the same time the office loses heat through the walls. We assume that the air properties in the office are homogeneous throughout the office.
a) How much water (in grams) would the air contain in a small one-person office of 20 cubic metres?
b) Suppose we measure a 30% relative humidity in this office. Calculate whether we can bring the relative humidity in the office up to the comfortable level by closing the air vents and putting a cup of water on the radiator?

Answer 1

(a)

$\varphi_1=\frac{\rho_1}{\rho_0}\cdot 100\%=50\%.$

From the table of dependence of saturated steam density on temperature $t=20°C, \rho_0=17.3 \frac{g}{m^3}.$

$\rho_1=\frac{\varphi_1\rho_0}{100}=\frac{50\cdot17.3}{100}=8.65 \frac{g}{m^3}.$

$m_1=\rho_1V=8.65\cdot20=173 \enspace g.$

(b)

$m_2=\rho_2V=\frac{\varphi_2\rho_0}{100}V=\frac{30}{100}\cdot17.3\cdot20=103.8\enspace g.$

$\Delta m=m_1-m_2=173-103.8=69.2\enspace g.$

It can be putted into a cup.