Question

In a certain solar house, energy from the Sun is stored in barrels filled with water. In a particular winter stretch of five cloudy days,$1.00\times {10}^6\text{kcal}$is needed to maintain the inside of the house at$22.0°\text{\hspace{0.17em}C}$. Assuming that the water in the barrels is at$50.0\text{ºC}$and that the water has a density of$1.00\times {10}^3\frac{\text{kg}}{{\text{m}}^{\text{3}}}$, what volume of water is required?

Short answer

The volume of water required is $35.7{\text{\hspace{0.17em}m}}^3$.

Step-by-step solution

The given data

a) Heat to be maintained inside the house for 5 days $Q=1.00\times {10}^6\frac{\text{kcal}}{\text{day}}$,

b) Temperature inside the house,$T=22°\text{C}$

c) Temperature of the water inside the barrels,$T=50°\text{C}$

d) Density of water,${\rho }_{water}={10}^3\frac{\text{kg}}{{\text{m}}^3}$

e) Specific heat of water,$C=1.00\frac{\text{kcal}}{\text{kg}°\text{C}}$

Determine the concept of heat absorption

The heat absorbed by the solar house is the required heat that is needed to be maintained inside the house. Thus, the amount of heat to be maintained should be constant. Now, the mass of the water considering the density given, we can get the volume of water inside the house at the maintained heat value.

Formulae:

The heat absorbed by a body due to its specific heat ( $C$), $Q=mC\Delta T$(i)

The volume of a body, $V=\frac{m\left(Mass\right)}{\rho \left(Density\right)}$ (ii)

Calculation of the volume of water

Substitute the value of mass from equation (i) in equation (ii) with the given data, the volume of the water maintained inside the barrels can be calculated as follows:

$\begin{array}{c}V=\frac{Q}{\rho C\Delta T}\\ =\frac{(1.00\times {10}^6\frac{\text{kcal}}{\text{day}})\left(\text{5\hspace{0.17em}days}\right)}{\left({10}^3\frac{\text{kg}}{{\text{m}}^3}\right)\left(1.00\frac{\text{kcal}}{\text{kg}°\text{C}}\right)(50°\text{C}-22°\text{C})}\\ =35.7{\text{\hspace{0.17em}m}}^3\end{array}$

Hence, the volume of water is $35.7{\text{\hspace{0.17em}m}}^3$.