Refrigeration performance metric: The heat-rejection factor (HRF) is defined as the ratio of heat rejected in the condenser to the refrigeration effect. Which expression is correct in terms of coefficient of performance (COP)?

Introduction / Context:
Designing condensers and cooling towers requires an estimate of total heat rejection. The heat-rejection factor (HRF) links the condenser duty to the evaporator load (refrigeration effect), using the coefficient of performance (COP) of the refrigeration machine.

Given Data / Assumptions:

• Refrigeration effect per unit mass = q_L.
• Work input per unit mass = w.
• Condenser heat rejected per unit mass = q_H = q_L + w.

Concept / Approach:
By definition, COP = q_L / w. The heat-rejection factor is HRF = q_H / q_L. Substitute q_H = q_L + w and express w in terms of COP to obtain HRF in a simple algebraic form.

Step-by-Step Derivation:

Verification / Alternative check:
For a COP of 4, HRF = 1 + 0.25 = 1.25. This matches practical rules of thumb that condenser load is about 1.2–1.3 times the cooling load for moderate COPs.

Why Other Options Are Wrong:

• 1 - 1/COP would predict condenser duty smaller than evaporator load, which is physically impossible.
• 1 + COP or the fractional forms offered do not arise from the energy balance q_H = q_L + w.

Common Pitfalls:
Confusing HRF with heat-rejection ratio expressed as a percent or confusing COP based on different definitions (cooling or heating mode).

Final Answer:
HRF = 1 + 1 / COP