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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