Question

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.

Short answer

Question: Calculate the heat transfer rate, the outlet temperatures of the hot and cold fluids, and the fouling factor based on the given problem and provided step-by-step solution.

Step-by-step solution

(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.)

(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.)

(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 \(Q\).)

(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 \(T_{c2}\).)

(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.)

Key concepts

Log Mean Temperature Difference

The Log Mean Temperature Difference (LMTD) is a crucial concept in the analysis of heat exchangers. It is particularly useful for determining the average temperature difference between two fluids in heat exchangers over a period of time. The LMTD helps in calculating the heat transfer rate efficiently. The formula for LMTD is given by:\[\Delta T_{lm} = \frac{(\Delta T_1 - \Delta T_2)}{\ln(\frac{\Delta T_1}{\Delta T_2})}\]Here, \(\Delta T_1\) is the temperature difference between the hot and cold fluids at one end, and \(\Delta T_2\) is the temperature difference at the other end. It's essential to note that the temperatures are taken simultaneously at the inlet and outlet of the heat exchanger. Calculating the accurate LMTD is vital because it significantly impacts the efficiency evaluation of the heat exchanger.

Overall Heat Transfer Coefficient

The Overall Heat Transfer Coefficient, denoted as \(U\), represents the heat exchanger's ability to conduct heat between fluids. It encompasses various resistances such as the material of the exchanger, fluid flow dynamics, and any surface fouling. The overall heat transfer is calculated in units of watts per square meter per degree Kelvin (W/m²⋅K). The main formula involving the Overall Heat Transfer Coefficient is:\[Q = U A \Delta T_{lm}\]where \(Q\) is the heat transfer rate, \(A\) is the surface area, and \(\Delta T_{lm}\) is the log mean temperature difference.
An accurate \(U\) value is critical for determining the heat exchanger's performance and efficiency, reflecting how effectively heat is being transferred from one fluid to the other.

Fouling Factor

The Fouling Factor is an additional resistance to heat transfer caused by the accumulation of unwanted deposits on the heat exchange surfaces. Over time, these deposits can significantly reduce efficiency. The fouling factor is calculated when there is a noticeable deviation in the overall heat transfer coefficient.The fouling factor \(R_f\) can be calculated using:\[R_{f} = \frac{1}{U_{2}A} - \frac{1}{U_{1}A}\]where \(U_1\) and \(U_2\) are the initial and reduced overall heat transfer coefficients, respectively, and \(A\) is the surface area of the heat exchanger.
Managing the fouling factor is vital as it can lead to reduced thermal efficiency, increased fuel consumption, and higher operational costs.

Energy Balance Equation

The Energy Balance Equation is crucial for understanding how energy is conserved in a heat exchanger. It essentially describes how the energy lost by the hot fluid is equal to the energy gained by the cold fluid under ideal conditions. This equation forms the basis for finding the outlet temperatures of fluids in heat exchangers.The equation can be written as:\[Q=mh c_{ph} (T_{h1}-T_{h2}) = mc c_{pc} (T_{c2}-T_{c1})\]where \(m\) is the mass flow rate, \(c_p\) is the specific heat capacity, \(T_{h1}\) and \(T_{h2}\) are the hot fluid inlet and outlet temperatures, while \(T_{c2}\) and \(T_{c1}\) are the cold fluid outlet and inlet temperatures, respectively. By solving this equation, one can determine the unknown outlet temperatures, balancing known inputs with the energy lost and gained by the fluids.
This balance ensures an accurate thermal design and operational reliability of the heat exchanger system.