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A counter-flow heat exchanger is stated to have an overall heat transfer coefficient of \(284 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) when operating at design and clean conditions. Hot fluid enters the tube side at \(93^{\circ} \mathrm{C}\) and exits at \(71^{\circ} \mathrm{C}\), while cold fluid enters the shell side at \(27^{\circ} \mathrm{C}\) and exits at \(38^{\circ} \mathrm{C}\). After a period of use, built-up scale in the heat exchanger gives a fouling factor of \(0.0004 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). If the surface area is \(93 \mathrm{~m}^{2}\), determine \((a)\) the rate of heat transfer in the heat exchanger and \((b)\) the mass flow rates of both hot and cold fluids. Assume both hot and cold fluids have a specific heat of \(4.2 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\).

Step by step solution

01

1. Calculate initial overall heat transfer coefficient (U_fouled)

To find the overall heat transfer coefficient when the heat exchanger has developed some fouling, we can use the following equation: $$U_fouled = \frac{1}{(1/U_{clean}) + R_{fouling}}$$ where \(U_{fouled}\) is the overall heat transfer coefficient in fouled conditions, \(U_{clean}\) is the overall heat transfer coefficient in clean conditions (given as 284 W/m²K), and \(R_{fouling}\) is the fouling factor (given as 0.0004 m²K/W). Plugging in given values, we can compute the \(U_{fouled}\).

02

2. Calculate the LMTD (Log Mean Temperature Difference)

To determine the LMTD of the temperature profile in the heat exchanger, we'll use the expression: $$LMTD = \frac{(T_{h1} -T_{c2}) - (T_{h2} - T_{c1})}{\ln[(T_{h1} - T_{c2}) / (T_{h2} - T_{c1})]}$$ where \(T_{h1}\) is the temperature of the hot fluid entering, \(T_{h2}\) is the temperature of the hot fluid leaving, \(T_{c1}\) is the cold fluid entering, and \(T_{c2}\) is the cold fluid leaving. Plugging in the given values, we can compute the LMTD.

03

3. Calculate the rate of heat transfer (Q)

Now that we have the LMTD and the overall heat transfer coefficients (both clean and fouled), we can calculate the rate of heat transfer using the equation: $$Q = U \cdot A \cdot LMTD$$ where \(Q\) represents the rate of heat transfer, \(U\) is the overall heat transfer coefficient (both clean and fouled), \(A\) is the surface area of the heat exchanger, and \(LMTD\) is the log mean temperature difference. We can compute the rate of heat transfer under both clean and fouled conditions.

04

4. Calculate the mass flow rate of hot and cold fluids (m_h, m_c)

To find the mass flow rates of the hot and cold fluids, we can use the equation \(\Delta T = (Q / mC_p)\) for each fluid, and rearrange the equation as: \(m = Q / (C_p\Delta T)\). The specific heat capacity (\(C_p\)) is given as 4200 J/kgK. For the hot fluid: $$m_h = \frac{Q}{C_H (T_{h1} - T_{h2})}$$ and for the cold fluid: $$m_c = \frac{Q}{C_C (T_{c2} - T_{c1})}$$. Solving for both mass flow rates, we can find \(m_h\) and \(m_c\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Overall Heat Transfer Coefficient

The overall heat transfer coefficient is a crucial factor in heat exchanger performance. It denotes the efficiency with which heat is transferred between fluids. It is symbolized by \(U\). This coefficient depends on factors such as the material's thermal conductivity, the fluid velocities, and the surface conditions of the heat exchanger.
During clean conditions, the coefficient has a higher value but as fouling develops, its value diminishes due to added thermal resistance. The formula to adjust for fouling is: \[U_{fouled} = \frac{1}{(1/U_{clean}) + R_{fouling}}\] Here, \(U_{clean}\) is the clean surface coefficient, and \(R_{fouling}\) is the fouling factor. Fouling mildly reduces \(U\), hence lowering effectiveness.

Fouling Factor

The fouling factor represents the resistance to heat transfer caused by deposits on heat exchanger surfaces. This occurs over time, especially in environments prone to scaling or muck formation. As fouling increases, the capacity of the heat exchanger to pass heat decreases, requiring more energy to accomplish the same level of heat transfer.
In calculations, it adds to the resistance in the heat transfer equation, thus lowering the effective heat transfer coefficient \(U\). It is a vital consideration in maintenance to ensure longevity and efficiency of heat exchangers.

Log Mean Temperature Difference

The Log Mean Temperature Difference (LMTD) is pivotal in quantifying the temperature driving force within a heat exchanger. Unlike simple averages, it considers the non-linear temperature gradient between the fluid streams. Calculating LMTD allows for more accurate determination of the heat exchanger’s performance: \[LMTD = \frac{(T_{h1} - T_{c2}) - (T_{h2} - T_{c1})}{\ln\left(\frac{T_{h1} - T_{c2}}{T_{h2} - T_{c1}}\right)}\] Where \(T_{h1}, T_{h2}, T_{c1},\) and \(T_{c2}\) denote inlet and outlet temperatures of the respective fluids. Accurate LMTD ensures precise energy transfer calculation.

Mass Flow Rate

The mass flow rate is an essential parameter indicating the amount of fluid mass moving through the heat exchanger per unit time. It affects the heat transfer rate significantly, given more mass can convey more heat energy. To determine the mass flow rates of hot \((m_h)\) and cold \((m_c)\) fluids, we'll use the energy balance: \[\Delta T = \frac{Q}{mC_p}\] Rearranging for \(m\) yields: \[m = \frac{Q}{C_p\Delta T}\] Where \(C_p\) is the specific heat. These calculations give insights into system dynamics and energy efficiency.

Specific Heat

Specific heat, denoted by \(C_p\), is an intrinsic material property that indicates how much energy a substance can store per unit mass and temperature change. It plays a vital role in heat exchanger calculations since it affects the heat transfer rate by influencing the \(\Delta T\) in the flow rate calculations: The specific heat value for both fluids in our exercise is given as \(4.2 \, \text{kJ/kg} \cdot \text{K}\). This knowledge helps engineers calculate energy needs and balance the exchanger's systems efficiently.