Heat Exchanger Effectiveness-NTU Method (Counterflow & Condenser)

Specify the idealized thermal model

Use steady-state heat transfer with constant heat-capacity rates and an overall heat-transfer coefficient U consistent with the chosen area A. Heat loss to the surroundings and axial conduction are neglected. Cmin and Cmax are the smaller and larger stream capacity rates, each mass flow times specific heat in W/K.

NTU=UA/Cmin and Cr=Cmin/Cmax. The counterflow relation is epsilon=(1-exp(-NTU(1-Cr)))/(1-Cr exp(-NTU(1-Cr))). At Cr=1 its limit is NTU/(1+NTU). Effectiveness is a heat-duty ratio, not an electrical efficiency or a guarantee of equipment performance.

For a single stream exchanging heat with an ideally constant-temperature phase-change side, the Cr=0 limit is 1-exp(-NTU). Real condensers and evaporators may include sensible-heating regions, temperature glide, pressure variation or maldistribution; do not apply that limit to an entire unit without checking the assumptions.

Compute NTU and the capacity ratio

B2 is U in W/(m² K), B3 area in m² and B4 Cmin in W/K. E2 returns NTU=2 for 500×12/3000. Zero U or zero area is allowed as the zero-transfer mathematical limit. Cmin must be strictly positive.

B5 is Cmax; E3 requires 0<Cmin<=Cmax. The initial ratio is 0.5. Reversed labels are reported instead of silently applying an invalid ratio.

=IF(COUNT(B2:B4)<>3,"Enter numeric U, area and Cmin",IF(OR(B2<0,B3<0,B4<=0),"Require U>=0, area>=0 and Cmin>0",B2*B3/B4))

Avoid loss of precision near balanced capacities

The direct counterflow expression produces 0/0 at Cr=1 and subtracts nearly equal numbers close to that limit. Helpers E4 and E5 evaluate x=NTU(1-Cr) and g(x)=(1-exp(-x))/x, using its limiting value at zero.

For 0<=x<0.0001, E5 uses 1-x/2+x²/6-x³/24+x⁴/120. This is a numerical Taylor approximation to the exponential expression; its first omitted term has magnitude at most x^5/720. For larger x it uses the exponential directly.

E6 then evaluates the algebraically equivalent NTU×g/(1+Cr×NTU×g). At Cr=1 it becomes the balanced limit automatically. The initial effectiveness is approximately 0.774600326; with Cmax=3000 it is 2/3. Outputs remain unrounded.

=IF(COUNT(E2:E5)<>4,"Check NTU and capacity inputs",E2*E5/(1+E3*E2*E5))

Compare the ideal isothermal phase-change limit

E7 uses the same NTU with Cr=0, independently of B5. It is a separate model comparison, not a claim that the two finite-capacity streams simultaneously have Cr=0. The initial value is approximately 0.864664717. A small-NTU polynomial avoids cancellation near zero.

=IF(NOT(ISNUMBER(E2)),E2,IF(E2<0.0001,E2*(1-E2/2+E2^2/6-E2^3/24+E2^4/120),1-EXP(-E2)))

Calculate counterflow heat duty

B6 and B7 are hot and cold inlet temperatures in °C. Their difference has the same numerical value in kelvins. E8 computes epsilon×Cmin×(Th,in−Tc,in) in watts. The initial 80°C and 20°C example gives about 139428.059 W.

Equal inlet temperatures give zero duty. A hot inlet below the cold inlet is rejected under this labeling convention. Outlet temperatures are not calculated here: Cmin alone does not say which stream is hot, so individual hot and cold capacity rates would also be needed.

=IF(NOT(ISNUMBER(E6)),E6,IF(COUNT(B6:B7)<>2,"Enter numeric inlet temperatures",IF(B6<B7,"Hot inlet must not be colder",E6*B4*(B6-B7))))

Further reading

• Caleffi — Thermal characteristics of heat exchangers
• Robert J. Ribando — Heat Exchanger Effectiveness-NTU Relations