Cooling and heating coils — by-pass factor for sensible heating: Given td1 = dry-bulb temperature of air entering the heating coil, td2 = dry-bulb temperature leaving the coil, and td3 = coil (surface) dry-bulb temperature, select the correct expression for by-pass factor (BPF) during sensible heating.

Introduction / Context:
The by-pass factor (BPF) quantifies the fraction of air that effectively does not contact (or equilibrate with) the coil surface. It is crucial for coil performance prediction in HVAC design.

Given Data / Assumptions:

• Sensible heating only (no moisture addition or removal).
• td3 represents the effective apparatus surface temperature for heating.
• Steady operation with uniform mixing downstream.

Concept / Approach:
For sensible heating, the leaving air temperature approaches the coil temperature. The by-pass factor is defined as the ratio of the temperature “gap” that remains after the coil to the initial gap at coil entry. For heating, the correct form is BPF = (td3 − td2) / (td3 − td1).

Step-by-Step Solution:

Verification / Alternative check:
If td2 equals td3 (perfect contact/equilibration), numerator is zero and BPF = 0, which matches the ideal of no by-pass. If no heating occurred (td2 = td1), BPF = 1, indicating total by-pass.

Why Other Options Are Wrong:

Forms that swap numerator/denominator or reverse temperatures correspond either to cooling cases or incorrect gaps and do not satisfy the limiting checks above.

Common Pitfalls:
Using the cooling-coil formula for a heating-coil case. Remember: heating uses (td3 − td2)/(td3 − td1); cooling uses (td2 − td3)/(td1 − td3).

Final Answer:

BPF = (td3 − td2) / (td3 − td1)