Question

Consider a condenser in which steam at a specified temperature is condensed by rejecting heat to the cooling water. If the heat transfer rate in the condenser and the temperature rise of the cooling water are known, explain how the rate of condensation of the steam and the mass flow rate of the cooling water can be determined. Also, explain how the total thermal resistance \(R\) of this condenser can be evaluated in this case.

Short answer

Question: Given a heat transfer rate of 5000 W, an enthalpy of vaporization of steam at 2.5 x 10^6 J/kg, a temperature rise of the cooling water at 10 K, a specific heat capacity of water at 4186 J/(kg·K), and a temperature difference between the steam and the inlet cooling water temperature of 50 K, determine the rate of condensation of the steam (in kg/s), the mass flow rate of the cooling water (in kg/s), and the total thermal resistance of the condenser (in K/W). Answer: 1. To find the rate of condensation of steam, use the formula: \(m_s = \dfrac{Q}{h_{fg}}\) \(m_s = \dfrac{5000}{2.5 \times 10^6}\) \(m_s \approx 0.002\) kg/s 2. To find the mass flow rate of the cooling water, use the formula: \(m_c = \dfrac{Q}{C_p \cdot \Delta T}\) \(m_c = \dfrac{5000}{4186 \times 10}\) \(m_c \approx 0.1194\) kg/s 3. To find the total thermal resistance of the condenser, use the formula: \(R = \dfrac{\Delta T_{total}}{Q}\) \(R = \dfrac{50}{5000}\) \(R \approx 0.01\) K/W Therefore, the rate of condensation of the steam is approximately 0.002 kg/s, the mass flow rate of the cooling water is approximately 0.1194 kg/s, and the total thermal resistance of the condenser is approximately 0.01 K/W.

Step-by-step solution

Calculate the rate of condensation of steam

Since the steam in the condenser loses heat to the cooling water and becomes liquid, we can use the heat transfer rate to find the rate of condensation of steam. We know that: Heat transfer rate \(Q = m_s \cdot h_{fg}\) where: \(Q\) = heat transfer rate [W] \(m_s\) = rate of condensation of steam [kg/s] \(h_{fg}\) = enthalpy of vaporization of steam [J/kg] We can rearrange the formula to find the rate of condensation of steam: \(m_s = \dfrac{Q}{h_{fg}}\)

Calculate the mass flow rate of the cooling water

We can find the mass flow rate of the cooling water by using the heat transfer rate and temperature rise of the cooling water: \(Q = m_c \cdot C_p \cdot \Delta T\) where: \(m_c\) = mass flow rate of the cooling water [kg/s] \(C_p\) = specific heat capacity of water [J/(kg·K)] \(\Delta T\) = temperature rise of the cooling water [K] We can rearrange the formula to find the mass flow rate of the cooling water: \(m_c = \dfrac{Q}{C_p \cdot \Delta T}\)

Evaluate the total thermal resistance of the condenser

We can evaluate the total thermal resistance \(R\) of the condenser by using the heat transfer rate and the temperature difference between the steam and the inlet cooling water temperature: \(Q = \dfrac{\Delta T_{total}}{R}\) where: \(\Delta T_{total}\) = temperature difference between the steam and the inlet cooling water temperature [K] We can rearrange the formula to find the total thermal resistance of the condenser: \(R = \dfrac{\Delta T_{total}}{Q}\) By using the given heat transfer rate and temperature rise of the cooling water, we can determine the rate of condensation of the steam and the mass flow rate of the cooling water. Additionally, knowing the temperature difference between the steam and the inlet cooling water temperature, we can evaluate the total thermal resistance of the condenser.