Question

Geothermal water $\left(c_{p}=1.03 \mathrm{Btu} / \mathrm{bm} \cdot{ }^{\circ} \mathrm{F}\right)$ is to be used as the heat source to supply heat to the hydronic heating system of a house at a rate of \(40 \mathrm{Btu} / \mathrm{s}\) in a double-pipe counterflow heat exchanger. Water $\left(c_{p}=1.0 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)$ is heated from \(140^{\circ} \mathrm{F}\) to \(200^{\circ} \mathrm{F}\) in the heat exchanger as the geothermal water is cooled from \(270^{\circ} \mathrm{F}\) to $180^{\circ} \mathrm{F}$. Determine the mass flow rate of each fluid and the total thermal resistance of this heat exchanger.

Short answer

Based on the given information and calculations, determine the mass flow rates of geothermal water and water, and the total thermal resistance of the heat exchanger. Mass flow rate of geothermal water: ≈ 0.3883 lbm/s Mass flow rate of water: 0.4 lbm/s Total thermal resistance of the heat exchanger: ≈ 1.33 °F·s/Btu

Step-by-step solution

Find the mass flow rates of geothermal water and water

We use the energy balance and the equation for heat transfer: Q = m * cp * ΔT For geothermal water (gw), Q_gw = m_gw * c_p_gw * (T_in_gw - T_out_gw) For water (w), Q_w = m_w * c_p_w * (T_out_w - T_in_w) Since the heat removed from the geothermal water is the same as the heat added to the water: Q_gw = Q_w Now let's calculate the mass flow rates: Q_required = 40 Btu/s (given) For the water: Q_w = m_w * c_p_w * (T_out_w - T_in_w) 40 = m_w * 1.0 * (200 - 140) m_w = 0.4 lbm/s For the geothermal water: Q_gw = m_gw * c_p_gw * (T_in_gw - T_out_gw) 40 = m_gw * 1.03 * (270 - 180) m_gw ≈ 0.3883 lbm/s

Calculate the total thermal resistance

To perform this step, we first need to calculate the overall heat transfer coefficient (UA) of the heat exchanger. The equation for the heat transfer rate in a double-pipe counterflow heat exchanger is as follows: Q = UA * ΔT_lm Where ΔT_lm is the logarithmic mean temperature difference, which can be calculated using the given inlet and outlet temperatures for both geothermal water and water: ΔT_1 = T_in_gw - T_out_w = 270 - 200 = 70°F ΔT_2 = T_out_gw - T_in_w = 180 - 140 = 40°F ΔT_lm = (ΔT_1 - ΔT_2) / ln(ΔT_1 / ΔT_2) ≈ 53.2°F Now we can find the overall heat transfer coefficient UA: Q_required = 40 Btu/s (given) ΔT_lm ≈ 53.2°F Q = UA * ΔT_lm UA ≈ 40 / 53.2 ≈ 0.7519 Btu/s·°F Now, we can calculate the total thermal resistance R_tot: UA = 1 / R_tot R_tot = 1 / UA ≈ 1.33 °F·s/Btu The mass flow rates of geothermal water and water are approximately 0.3883 lbm/s and 0.4 lbm/s, respectively, and the total thermal resistance of the heat exchanger is approximately 1.33 °F·s/Btu.