Heat exchanger effectiveness using inlet/outlet temperatures For a heat exchanger where the cold fluid has the lower heat capacity rate, and temperatures are: cold in t_c1, cold out t_c2; hot in t_h1, hot out t_h2. What is the effectiveness ε expressed in terms of these temperatures?

Introduction / Context:
Effectiveness–NTU analysis provides a robust way to rate heat exchangers without detailed area or U values. This question focuses on the temperature-form definition when the cold stream has the minimum heat capacity rate.

Given Data / Assumptions:

• Cold fluid is the minimum-capacity stream (C_min).
• Steady operation with no heat losses to surroundings.
• Single-phase sensible heating/cooling.

Concept / Approach:
Effectiveness is defined as ε = q_actual / q_max. With cold fluid as C_min, the maximum possible temperature rise of the cold stream is to reach the hot inlet temperature t_h1, giving q_max = C_min * (t_h1 − t_c1). The actual heat gained by the cold stream is q_actual = C_cold * (t_c2 − t_c1). Because C_min = C_cold here, the C terms cancel in the ratio.

Step-by-Step Solution:
Write ε = q_actual / q_max.q_actual = C_cold * (t_c2 − t_c1).q_max = C_min * (t_h1 − t_c1) = C_cold * (t_h1 − t_c1).Therefore ε = (t_c2 − t_c1) / (t_h1 − t_c1).

Verification / Alternative check:
Check bounds: if t_c2 → t_h1, ε → 1; if no heating (t_c2 = t_c1), ε = 0. Both are physically consistent.

Why Other Options Are Wrong:

• (t_h1 − t_h2)/(t_h1 − t_c1): would be correct only if hot stream were C_min.
• Other forms mix outlet temperatures inconsistently with the definition of q_max.

Common Pitfalls:
Using the wrong stream as C_min; forgetting that effectiveness is independent of flow arrangement in this temperature-form expression, while ε–NTU relationships depend on configuration.

Final Answer:
ε = (t_c2 − t_c1) / (t_h1 − t_c1)