Question

Water is heated in an insulated, constant diameter tube by a \(7-\mathrm{kW}\) electric resistance heater. If the water enters the heater steadily at \(15^{\circ} \mathrm{C}\) and leaves at \(70^{\circ} \mathrm{C}\), determine the mass flow rate of the water.

Short answer

Answer: The mass flow rate of the water is approximately \(0.0304\,\mathrm{kg/s}\).

Step-by-step solution

List the given information

Electric power of the heater: \(P_{heater} = 7\,\mathrm{kW}\) Temperature of water entering the heater: \(T_{in} = 15^{\circ}\mathrm{C}\) Temperature of water leaving the heater: \(T_{out} = 70^{\circ}\mathrm{C}\) Specific heat capacity of water: \(c_{water} = 4.186\,\mathrm{kJ/kg \cdot K}\)

Convert the heater power to watts

We need to convert the heater power from kilowatts (kW) to watts (W) since the specific heat is in terms of joules, and 1 W equals 1 J/s. We do this by multiplying the given value by 1,000. \(P_{heater} = 7\,000\,\mathrm{W}\)

Calculate the change in temperature

Next, we find the change in temperature \(\Delta T\) by subtracting the initial temperature from the final temperature. \(\Delta T = T_{out} - T_{in} = 70^{\circ}\mathrm{C} - 15^{\circ}\mathrm{C} = 55^{\circ}\mathrm{C}\)

Rearrange the power formula to find the mass flow rate

The power formula is given by \(P = mcΔT\). We need to rearrange it to isolate the mass flow rate, \(m\). To do this, we divide both sides by \(c\Delta T\): \(m = \frac{P_{heater}}{c_{water} \cdot \Delta T}\)

Substitute the values into the rearranged formula

Now, we can substitute the values of \(P_{heater}\), \(c_{water}\), and \(\Delta T\) into the equation to find the mass flow rate of the water: \(m = \frac{7\,000\,\mathrm{W}}{4.186\,\mathrm{kJ/kg \cdot K} \cdot 55^{\circ}\mathrm{C}}\)

Convert the specific heat capacity to the correct units

The specific heat of water has units of kJ/(kg⋅K), but we need to convert it to J/(kg⋅K) to match the units of the heater power. To do this, we multiply by 1,000: \(c_{water} = 4.186\,\mathrm{kJ/kg \cdot K} \times 1\,000 = 4\,186\,\mathrm{J/kg \cdot K}\)

Calculate the mass flow rate

Substitute the correct units for specific heat capacity and then calculate the mass flow rate: \(m = \frac{7\,000\,\mathrm{W}}{4\,186\,\mathrm{J/kg \cdot K} \cdot 55^{\circ}\mathrm{C}} = \frac{7\,000\,\mathrm{J/s}}{230\,230\,\mathrm{J/kg}} \approx 0.0304\, \mathrm{kg/s}\) Therefore, the mass flow rate of the water is approximately \(0.0304\,\mathrm{kg/s}\).