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 is 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.

Heat transfer is the process of transmission of energy, in this case, thermal energy from a high-temperature substance to a lower temperature one. In a condenser, steam transfers heat to the cooling water, resulting in a decrease in its temperature (steam gets condensed) and an increase in the cooling water's temperature. The thermal resistance of a material is a measure of how effectively it can resist the flow of heat. In this case, we are interested in the total thermal resistance R of the condenser.

We have the following information: 1. Heat transfer rate in the condenser 2. Temperature rise of the cooling water Our goal is to determine the rate of condensation of the steam and the mass flow rate of the cooling water. Then, the total thermal resistance R of the condenser will be evaluated.

Let's denote the heat transfer rate as \(Q\), the mass flow rate of steam as \(\dot{m}_s\), the initial temperature of the steam as \(T_{s_i}\), and the final temperature of the steam as \(T_{s_f}\) after condensation. We know that the heat transfer from the steam to the cooling water causes the steam to condense, which means the change in enthalpy is equal to the heat transfer: \(Q = \dot{m}_s (h_{s_f} - h_{s_i})\) where \(h_{s_i}\) and \(h_{s_f}\) are the enthalpies of the steam before and after condensation, respectively. From the given information and knowing the enthalpy of steam at the specified temperature, the rate of condensation of steam \(\dot{m}_s\) can be determined by rearranging the equation: \(\dot{m}_s = \dfrac{Q}{h_{s_f} - h_{s_i}}\)

Let's denote the mass flow rate of the cooling water as \(\dot{m}_w\), the initial temperature of the cooling water as \(T_{w_i}\), and the final temperature of the cooling water as \(T_{w_f}\) after receiving heat from the steam. The specific heat capacity of the cooling water is \(c_w\). The heat transfer to the cooling water is given by: \(Q = \dot{m}_w c_w (T_{w_f} - T_{w_i})\) We know the heat transfer rate \(Q\) and the temperature difference \((T_{w_f} - T_{w_i})\) from the given information. Therefore, we can determine the mass flow rate of the cooling water by rearranging the equation: \(\dot{m}_w = \dfrac{Q}{c_w (T_{w_f} - T_{w_i})}\)

The total thermal resistance R of the condenser can be obtained using the overall heat transfer equation as follows: \(Q = U A \Delta{T}_L\) where \(U\) is the overall heat transfer coefficient, \(A\) is the heat transfer area, and \(\Delta{T}_L\) is the logarithmic mean temperature difference between the steam and cooling water. Using the given heat transfer rate \(Q\), we can determine the total thermal resistance R as: \(R = \dfrac{1}{U A} = \dfrac{\Delta{T}_L}{Q}\)