Question

Consider a counter-flow heat exchanger with the inlet temperatures of two fluids (1 and 2) being \(T_1 = 300\) K and \(T_2 = 350\) K. The heat capacity rates of the two fluids are \(C_1 = 1000\) W/K and \(C_2 = 400\) W/K, and the effectiveness of the heat exchanger is 0.5. The actual heat transfer rate is (in kW)

• A. 12
• B. 15
• C. 18
• D. 10

Hint:

Always be careful with units when solving heat capacity problems. Remember: \[ \text{Watts (W)} \rightarrow \text{kiloWatts (kW)} \text{ divide by } 1000. \] Crucially, always evaluate \(C_{\text{min}}\) to calculate maximum possible heat transfer, because the fluid with the smaller heat capacity rate will limit the overall heat exchange process by reaching its thermodynamic temperature limit first.

Answer 1

The Correct Option is D

Concept: According to the Effectiveness-NTU method for heat exchanger analysis, the thermal effectiveness (\(\varepsilon\)) is defined as the ratio of the actual heat transfer rate (\(Q_{\text{actual}}\)) to the maximum thermodynamically possible heat transfer rate (\(Q_{\text{max}}\)): \[ \varepsilon = \frac{Q_{\text{actual}}}{Q_{\text{max}}} \quad \implies \quad Q_{\text{actual}} = \varepsilon \cdot Q_{\text{max}} \] The maximum heat transfer capacity rate is bounded by the fluid possessing the minimum heat capacity rate (\(C_{\text{min}}\)), combined with the maximum temperature difference existing within the system boundaries: \[ Q_{\text{max}} = C_{\text{min}} \left( T_{\text{hot, in}} - T_{\text{cold, in}} \right) \]

Step 1: Identify hot and cold fluid inlet conditions
Based on the provided numerical data, Fluid 2 enters at a higher thermal state than Fluid 1: \[ T_{\text{hot, in}} = T_2 = 350\text{ K} \] \[ T_{\text{cold, in}} = T_1 = 300\text{ K} \] Thus, the total maximum temperature differential spanning across the exchanger is: \[ \Delta T_{\text{max}} = T_{\text{hot, in}} - T_{\text{cold, in}} = 350\text{ K} - 300\text{ K} = 50\text{ K} \]

Step 2: Determine the minimum heat capacity rate (\(C_{\text{min}}\))
The heat capacity rates given for each respective stream are: \[ C_1 = 1000\text{ W/K} \] \[ C_2 = 400\text{ W/K} \] Comparing these rates explicitly: \[ C_{\text{min}} = \min(C_1, C_2) = \min(1000, 400) = 400\text{ W/K} \]

Step 3: Calculate the maximum possible heat transfer rate
Using the definition of maximum performance capacity: \[ Q_{\text{max}} = C_{\text{min}} \cdot \Delta T_{\text{max}} \] \[ Q_{\text{max}} = 400\text{ W/K} \times 50\text{ K} = 20000\text{ W} = 20\text{ kW} \]

Step 4: Determine the actual heat transfer rate using effectiveness
Given that the structural thermal effectiveness parameter is \(\varepsilon = 0.5\), we can solve for the actual heat transmission: \[ Q_{\text{actual}} = \varepsilon \cdot Q_{\text{max}} \] \[ Q_{\text{actual}} = 0.5 \times 20\text{ kW} = 10\text{ kW} \] Hence, the system transfers a total thermal rate of \(10\text{ kW}\), matching option (4).