Question

Hot water at \(60^{\circ} \mathrm{C}\) is cooled to \(36^{\circ} \mathrm{C}\) through the tube side of a 1-shell pass and 2-tube passes heat exchanger. The coolant is also a water stream, for which the inlet and outlet temperatures are \(7^{\circ} \mathrm{C}\) and \(31^{\circ} \mathrm{C}\), respectively. The overall heat transfer coefficient and the heat transfer area are \(950 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(15 \mathrm{~m}^{2}\), respectively. Calculate the mass flow rates of hot and cold water streams in steady operation.

Short answer

Answer: The mass flow rate of hot water (m_h) is 15.579 kg/s, and the mass flow rate of cold water (m_c) is 18.096 kg/s.

Step-by-step solution

Calculate the log mean temperature difference (ΔT_lm)

To find the log mean temperature difference, we use the given temperatures of the hot and cold water streams. We have the following temperatures: - Hot water inlet: 60°C - Hot water outlet: 36°C - Cold water inlet: 7°C - Cold water outlet: 31°C We can calculate the log mean temperature difference using the formula: ΔT_lm = (ΔT_1 - ΔT_2) / ln(ΔT_1 / ΔT_2), where ΔT_1 and ΔT_2 are the temperature differences at the two ends of the heat exchanger. Here, ΔT_1 = 60 - 7 = 53°C, and ΔT_2 = 36 - 31 = 5°C. So, we get: ΔT_lm = (53 - 5) / ln(53 / 5) ≈ 21.65°C.

Calculate the heat transfer rate (Q)

Now, we can find the heat transfer rate (Q) using the formula: Q = U * A * ΔT_lm Here, U = 950 W/m²·K, A = 15 m², and ΔT_lm = 21.65°C. So, we get: Q = 950 * 15 * 21.65 ≈ 308875 W = 308.875 kJ/s.

Calculate the mass flow rate of hot water (m_h)

To find the mass flow rate of hot water, we can use the energy balance equation for the hot water stream: Q = m_h * Cp * (T_inlet - T_outlet) Here, Cp ≈ 4.18 kJ/kg·K, T_inlet = 60°C, and T_outlet = 36°C. Rearranging the equation for m_h, we get: m_h = Q / (Cp * (T_inlet - T_outlet)) Now substituting the values, we get: m_h = 308.875 / (4.18 * (60 - 36)) ≈ 15.579 kg/s.

Calculate the mass flow rate of cold water (m_c)

Similarly, to find the mass flow rate of cold water, we can use the energy balance equation for the cold water stream: Q = m_c * Cp * (T_outlet - T_inlet) Here, T_outlet = 31°C and T_inlet = 7°C. Rearranging the equation for m_c, we get: m_c = Q / (Cp * (T_outlet - T_inlet)) Now substituting the values, we get: m_c = 308.875 / (4.18 * (31 - 7)) ≈ 18.096 kg/s. Finally, we have calculated the mass flow rates for both hot and cold water streams: - Mass flow rate of hot water (m_h) = 15.579 kg/s - Mass flow rate of cold water (m_c) = 18.096 kg/s

Key concepts

Log Mean Temperature Difference

The log mean temperature difference (LMTD) is crucial in heat exchanger calculations. It represents the effective temperature driving force between hot and cold fluids across the heat exchanger. To understand it better, think of it as a way to find an average temperature difference when these differences aren't constant, especially across heat exchangers like the one in this exercise.
In our heat exchanger, the hot water is cooled from 60°C to 36°C, while the cold water warms from 7°C to 31°C. These temperature changes show different rates and hence different temperature differences across the heat exchanger. Using the formula \( \Delta T_{\text{lm}} = \frac{\Delta T_1 - \Delta T_2}{\ln\left(\frac{\Delta T_1}{\Delta T_2}\right)} \), where \( \Delta T_1 = 53°C \) and \( \Delta T_2 = 5°C \), the LMTD is calculated as approximately 21.65°C.
Remember, using LMTD helps simplify the analysis by giving a single driving force value, enabling easier computations of the heat exchanger's performance.

Overall Heat Transfer Coefficient

The overall heat transfer coefficient, denoted as \(U\), is a measure of a heat exchanger's efficiency in transferring heat between fluids. It encompasses various resistances to heat flow, including the convective heat transfer within the fluids and conductive resistance through the heat exchanger material.
In our example, \(U\) is given as 950 W/m²·K. This value combines the properties of the materials and fluids involved to show how effectively the heat exchanger transfers heat per unit area for each degree of temperature difference. The effectiveness of \(U\) directly impacts the heat transfer rate \(Q\), calculated using \(Q = U \times A \times \Delta T_{\text{lm}}\), where \(A\) is the heat transfer area (15 m²) and \( \Delta T_{\text{lm}}\) is the log mean temperature difference.
By determining \(Q\) as approximately 308,875 W, we see how well our heat exchanger performs in transferring energy from hot to cold water, driven by the overall heat transfer coefficient.

Mass Flow Rate Calculation

The mass flow rate calculation is vital for understanding the amounts of fluid moving through a heat exchanger per unit of time. It connects the energy exchanged between streams and their specific heat capacity. In essence, it shows how much fluid is needed to achieve the desired heat exchange results.
For the hot water stream, the mass flow rate \(m_h\) was calculated using the formula \(Q = m_h \times Cp \times (T_{\text{inlet}} - T_{\text{outlet}})\). With \(Cp = 4.18\, \text{kJ/kg·K}\), \(T_{\text{inlet}} = 60°C\), and \(T_{\text{outlet}} = 36°C\), \(m_h\) is approximately 15.579 kg/s.
Similarly, the mass flow rate for cold water \(m_c\) uses a similar equation, \(Q = m_c \times Cp \times (T_{\text{outlet}} - T_{\text{inlet}})\), with \(T_{\text{inlet}} = 7°C\) and \(T_{\text{outlet}} = 31°C\). This gives a mass flow rate \(m_c\) of approximately 18.096 kg/s.
These calculations inform us how much of each fluid must flow to achieve the specified cooling or heating effects, making them essential for designing and understanding heat exchanger operations.