Problem — electric boiler

Problem

Estimate the rated power of an electric boiler to heat $150\ \text{L}$ of water from $15\ ^\circ$C to $40\ ^\circ$C in $30$ minutes. Consider an efficiency of $90\%$ and use $c_\text{water} = 4186\ \text{J/(kg}\cdot\text{K)}$. Compare the result with the typical power ratings of commercial boilers and discuss the need for a dedicated electrical line.

Solution

Useful heat required — $150\ \text{L}$ of water have mass $m = 150\ \text{kg}$; the temperature rise is $\Delta T = 40 - 15 = 25\ \text{K}$: $Q = m\,c\,\Delta T = 150 \cdot 4186 \cdot 25 \approx 1{,}57\cdot 10^{7}\ \text{J}$

Electrical energy absorbed — with efficiency $\eta = 0{,}90$ (part of the heat is lost), the energy that must be supplied is greater: $E = \frac{Q}{\eta} = \frac{1{,}57\cdot 10^{7}}{0{,}90} \approx 1{,}74\cdot 10^{7}\ \text{J}$

Rated power — in $t = 30\ \text{min} = 1800\ \text{s}$: $P = \frac{E}{t} = \frac{1{,}74\cdot 10^{7}\ \text{J}}{1800\ \text{s}} \approx 9{,}7\cdot 10^{3}\ \text{W} \approx 9{,}7\ \text{kW}$

Comparison and dedicated line — commercial electric boilers typically range between $2$ and $10\ \text{kW}$: the estimated value is realistic for a high-end model. At $230\ \text{V}$ a power of $\sim 10\ \text{kW}$ means a current $i = P/V \approx 42\ \text{A}$, well beyond the $16\ \text{A}$ of a standard domestic socket and even beyond the $\sim 25\text{–}30\ \text{A}$ of the residential meter. A dedicated electrical line is therefore needed, with large cross-section cables and its own protection — or, alternatively, longer heating times (lower power) or three-phase supply.

$\ev{P_\text{nom} \sim 9{,}7\ \text{kW}}$

Collegamenti

Argomenti: Circuiti elettrici Concetti: Effetto Joule Competenze: Stime di Fermi