# Condenser Area Derivation

For a configuration like the cross-flow Condenser, the area on both refrigerant and air sides is proportional to a parameter , the length fraction for a given circuit. In addition, the air mass flow rate is proportional to the parameter . As a result, independent of whether the minimum capacitance rate is on the air- or refrigerant-side, the same result for the area fraction is obtained, as shown from the derivation below.

For this derivation, the inlet temperatures of both streams are known, and the outlet stream of the refrigerant is known. In addition, the mass flow rates of refrigerant and air are known. Therefore, the actual amount of heat transfer is also known.

(1)

## Minimum capacitance rate on air side

If on air side:

(2)

cancels out, leaving independent of . Energy balance yields

(3)

(4)

(5)

LHS is constant, call it . The minimum capacitance rate is on the air side ( ), cross-flow with mixed (ref.) and unmixed (air) yields

(6)

Now solve for

(7)

(8)

(9)

(10)

Coming back to the definition of as the ratio of capacitance rates, you can get from

(11)

and since is already known, you obtain

(12)

(13)

## Minimum capacitance rate on refrigerant side

If on refrigerant side:

(14)

(15)

(16)

Energy balance yields

(17)

(18)

(19)

Right-hand-side is also equal to from above. Effectiveness with mixed (ref.) and unmixed (air) yields

(20)

(21)

(22)

(23)

(24)

(25)

Thus both assuming that the minimum capacitance rate is on the air- or refrigerant-sides yields exactly the same solution, which conveniently allows for an explicit solution independent of whether the air-side is the limiting capacitance rate or not