C

Step 1: Identify the parameters for calculations
We need to heat 10 kg of water from 20°C to 80°C. The specific heat of water is approximately 1 cal/g°C, and the latent heat of vaporization is approximately 540 cal/g.

Step 2: Calculate the heat required to heat the water
The formula for heat required (Q) is:
$Q = m imes c imes riangle T$
Where:
- $m$ = mass (in grams),
- $c$ = specific heat,
- $ riangle T$ = change in temperature.

Converting the mass of water: $10 ext{ kg} = 10000 ext{ g}$.

Calculate $ riangle T$:
$$ riangle T = 80°C - 20°C = 60°C$$

Now calculate Q:
$$Q = 10000 ext{ g} imes 1 rac{ ext{cal}}{ ext{g°C}} imes 60°C = 600000 ext{ cal}$$

Step 3: Find the mass of steam required
The heat provided by steam condensing (Q) is:
$$Q = m_{steam} imes L$$
Where:
- $m_{steam}$ = mass of steam (in grams),
- $L$ = latent heat of vaporization.

Setting $Q$ equal to the heat required for heating the water gives us the equation:
$$600000 ext{ cal} = m_{steam} imes 540 ext{ cal/g}$$

Solving for $m_{steam}$:
$$m_{steam} = \frac{600000 ext{ cal}}{540 ext{ cal/g}} \approx 1111.11 ext{ g}$$

Convert this to kg:
$$m_{steam} \approx 1.11 ext{ kg}$$

Step 4: Final Result
To summarize, the mass of steam required is approximately 1.11 kg per hour. Therefore, the correct answer is C.