Question

The effectiveness of a heat exchanger is defined as the ratio of the actual heat transfer rate to

• A. The maximum possible heat-transfer rate
• B. The minimum possible heat-transfer rate
• C. The overall heat-transfer coefficient
• D. The area of the heat exchanger

Hint:

The effectiveness method is particularly useful when the outlet temperatures of the fluids are not known, making the Log Mean Temperature Difference (LMTD) method difficult to apply directly. The effectiveness depends on the heat capacity ratio \( (C_{min}/C_{max}) \) and the Number of Transfer Units (NTU).

Answer 1

The Correct Option is A

Step 1: Understand the concept of heat exchanger effectiveness.
The effectiveness \( (\epsilon) \) of a heat exchanger is a dimensionless parameter that indicates how close the actual heat transfer rate is to the thermodynamically maximum possible heat transfer rate under the given conditions. It provides a measure of the heat exchanger's performance. Step 2: Recall the definition of effectiveness.
The effectiveness of a heat exchanger is defined as the ratio of the actual rate of heat transfer to the maximum possible rate of heat transfer. Mathematically, this is expressed as: $$\epsilon = \frac{Q_{actual}}{Q_{maximum}}$$ Step 3: Understand the terms in the definition.
\( Q_{actual} \) is the actual rate of heat transfer occurring in the heat exchanger. This can be calculated using the heat capacity rates and the temperature differences of the hot and cold fluids. For example, \( Q_{actual} = C_h (T_{h,i} - T_{h,o}) = C_c (T_{c,o} - T_{c,i}) \), where \( C \) represents the heat capacity rate (\( m \cdot c_p \)), and \( i \) and \( o \) denote inlet and outlet conditions for the hot (\( h \)) and cold (\( c \)) fluids.
\( Q_{maximum} \) is the thermodynamically maximum possible rate of heat transfer. This would occur in a counter-flow heat exchanger of infinite length. The maximum heat transfer is limited by the fluid with the minimum heat capacity rate \( (C_{min}) \) and the maximum temperature difference available in the heat exchanger, which is the difference between the inlet temperatures of the hot and cold fluids \( (T_{h,i} - T_{c,i}) \). Therefore,
$$Q_{maximum} = C_{min} (T_{h,i} - T_{c,i})$$
where \( C_{min} = \min(C_h, C_c) \).
Step 4: Combine the expressions to understand the ratio.
The effectiveness is then:
$$\epsilon = \frac{Q_{actual}}{C_{min} (T_{h,i} - T_{c,i})}$$ From this definition, it is clear that the effectiveness is the ratio of the actual heat transfer rate to the maximum possible heat transfer rate.