Capacity Rate Ratio (Cr)

C_r = \frac{\dot{m}_{min} c_{min}}{\dot{m}_{max} c_{max}}

Worked example: Air 1206 W/K over water 3348.8 W/K → Cr 0.3601 — press Try an example to run it live, then adjust anything.

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Capacity Rate Ratio (Cr) explained

Heat capacity rate, C = ṁcₚ in watts per kelvin, is how much heat a stream absorbs for each degree it warms. Divide the smaller by the larger and you get Cr, which by construction runs from 0 to 1 and controls how effectiveness responds to size. Example: 1.2 kg/s of air (cₚ ≈ 1005) is 1206 W/K; 0.8 kg/s of water (cₚ ≈ 4186) is 3349 W/K; Cr = 0.36, and the air — despite the higher flow — is the limiting stream, because water carries four times the heat per kilogram per degree.

The two ends of the range are the interesting ones. Cr = 0 means one stream's capacity rate is effectively infinite, which is exactly what happens when a fluid boils or condenses: it absorbs heat at constant temperature, and every exchanger arrangement — counterflow, parallel, crossflow — collapses to the same $\varepsilon = 1 - e^{-\mathrm{NTU}}$ . Cr = 1 is the balanced exchanger, hardest to make effective, and the case where counterflow's advantage over parallel flow is largest. The trap is bookkeeping: identify $C_{\mathrm{min}}$ from ṁcₚ, not from flow rate alone. Steam-to-water and refrigerant-to-air units are Cr = 0 problems no matter what the flow meters read.

Capacity Rate Ratio (Cr) formula

C_r = \frac{\dot{m}_{min} c_{min}}{\dot{m}_{max} c_{max}}

Where

$C_r$ = Capacity rate ratio
$\dot{m}_{min}$ = Minimum stream mass flow (kg/h)
$c_{min}$ = Minimum stream specific heat (J/(kg·K))
$\dot{m}_{max}$ = Maximum stream mass flow (kg/h)
$c_{max}$ = Maximum stream specific heat (J/(kg·K))

Missing one of these? Work it out first, then come back

Capacity rate ratio — Otto Cycle Efficiency (Compression Ratio), Brayton Cycle Efficiency (Pressure Ratio)
Minimum stream mass flow — Number of Transfer Units (NTU), Maximum Possible Heat Transfer (Qmax)
Minimum stream specific heat — Number of Transfer Units (NTU), Maximum Possible Heat Transfer (Qmax)
Maximum stream mass flow — Number of Transfer Units (NTU), Maximum Possible Heat Transfer (Qmax)
Maximum stream specific heat — Number of Transfer Units (NTU), Maximum Possible Heat Transfer (Qmax)

Part of this formula set

The effectiveness-NTU method

5 formulasSizing or rating a heat exchanger without knowing the outlet temperatures — capacity rates, NTU, effectiveness and the maximum possible duty.