A counter-flow heat exchanger is attached to a stenter for waste heat recovery.
Given data: \[\begin{array}{rl} \bullet & \text{Ambient temperature, \( T_1 = 30^\circ C \)} \\ \bullet & \text{Temperature of exhaust from stenter, \( T_2 = 150^\circ C \)} \\ \bullet & \text{Temperature of exhaust at exit of heat exchanger, \( T_3 = 100^\circ C \)} \\ \bullet & \text{Specific heat of exhaust gas, \( C_p = 0.42 \, \text{kcal} \cdot \text{deg}^{-1} \cdot \text{kg}^{-1} \)} \\ \bullet & \text{Specific heat of air, \( C_a = 0.24 \, \text{kcal} \cdot \text{deg}^{-1} \cdot \text{kg}^{-1} \)} \\ \end{array}\]
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In a counter-flow heat exchanger, energy balance equations can be used to calculate the outlet temperature of one fluid, given the inlet temperatures, specific heats, and the heat exchanged.
Given the following data:
\[\begin{array}{rl} \bullet & \text{Ambient temperature: \( 30^\circ C \)} \\ \bullet & \text{Temperature of exhaust from stenter: \( 150^\circ C \)} \\ \bullet & \text{Temperature of exhaust at exit of heat exchanger: \( 100^\circ C \)} \\ \bullet & \text{Specific heat of exhaust: \( 0.42 \, \text{cal/g/}^\circ \text{C} \)} \\ \bullet & \text{Specific heat of air: \( 0.24 \, \text{cal/g/}^\circ \text{C} \)} \\ \end{array}\]
At steady state, if the mass flow rates of the exhaust gas and the incoming air are the same, and assuming heat loss as zero, we can use the formula for the heat exchanger in counter-flow configuration:
\[
Q = m \cdot c \cdot \Delta T = m \cdot c_e \cdot (T_{\text{exhaust in}} - T_{\text{exhaust out}}) = m \cdot c_a \cdot (T_{\text{air out}} - T_{\text{air in}}).
\]
By equating both sides and solving for \( T_{\text{air out}} \), the temperature of the air at the exit of the heat exchanger, we find:
\[
T_{\text{air out}} \approx 117.0^\circ C.
\]
Thus, the temperature of the air at the exit of the heat exchanger is \( 117.0^\circ C \).
