Hot water \(\left(c_{p h}=4188 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) with mass flow rate of \(2.5 \mathrm{~kg} / \mathrm{s}\) at \(100^{\circ} \mathrm{C}\) enters a thin-walled concentric tube counterflow heat exchanger with a surface area of \(23 \mathrm{~m}^{2}\) and an overall heat transfer coefficient of \(1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Cold water \(\left(c_{p c}=4178 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) with mass flow rate of \(5 \mathrm{~kg} / \mathrm{s}\) enters the heat exchanger at \(20^{\circ} \mathrm{C}\), determine \((a)\) the heat transfer rate for the heat exchanger and \((b)\) the outlet temperatures of the cold and hot fluids. After a period of operation, the overall heat transfer coefficient is reduced to \(500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine (c) the fouling factor that caused the reduction in the overall heat transfer coefficient.

Step by step solution

01

(Step 1: Find the heat transfer rate)

(To determine the heat transfer rate, we'll make use of the given equations and values. The heat transfer rate (Q) can be calculated using the equation: \(Q=UA\Delta T_{lm}\), where \(U\) is the overall heat transfer coefficient, \(A\) is the surface area, and \(\Delta T_{lm}\) is the log mean temperature difference. We'll first find the log mean temperature difference.)

02

(Step 2: Calculate the Log Mean Temperature Difference)

(The log mean temperature difference (\(\Delta T_{lm}\)) can be calculated using the equation \(\Delta T_{lm}=\frac{(\Delta T_1-\Delta T_2)}{\ln(\frac{\Delta T_1}{\Delta T_2})}\), where \(\Delta T_1=T_{h1}-T_{c1}\) and \(\Delta T_2=T_{h2}-T_{c2}\). The inlet and outlet temperatures of the hot and cold fluids are given, so we can find the values for \(\Delta T_1\) and \(\Delta T_2\). Note that at this stage, we do not know the outlet temperatures of the cold and hot fluids, so we will work with \(T_{h2}\) and \(T_{c2}\) instead.)

03

(Step 3: Calculate the Heat Transfer Rate)

(Now that we have the log mean temperature difference, we can calculate the heat transfer rate using the equation \(Q=UA\Delta T_{lm}\). Considering the given overall heat transfer coefficient and surface area, we can find the value for \(Q\).)

04

(Step 4: Determine the Outlet Temperatures of Hot and Cold Fluids)

(To find the outlet temperatures, we'll use the heat transfer rate found in step 3 and the respective heat capacities and mass flow rates of the hot and cold fluids. Using the energy balance equation \(Q=mh c_{ph} (T_{h1}-T_{h2})=mc c_{pc} (T_{c2}-T_{c1})\), we can solve for the outlet temperatures of the hot and cold fluids, \(T_{h2}\) and \(T_{c2}\).)

05

(Step 5: Calculate the Fouling Factor)

(After a period of operation, the overall heat transfer coefficient is reduced from 1000 to 500 watts per square meter-kelvin. We can use the equation \(R_{f}=\frac{1}{U_{2}A}-\frac{1}{U_{1}A}\) to solve for the fouling factor, \(R_{f}\), where \(U_1\) is the initial overall heat transfer coefficient, \(U_2\) is the reduced overall heat transfer coefficient, and \(A\) is the surface area. Calculate the fouling factor using the given values.)

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