E3 · HVAC Equipment and Components

Foundation lesson · 18 minute study session

Match the heat exchanger to its terminal temperatures.

Use flow arrangement, capacity rate and area basis to select a defensible heat-transfer method.

By the end of this lesson

Pair the correct terminal temperature differences for counterflow.
Apply the logarithmic mean and its equal-difference limit.
Distinguish an LMTD calculation from effectiveness and NTU methods.

Two streams share one duty.

For a steady exchanger with negligible external loss, heat removed from the hot stream equals heat gained by the cold stream. Define capacity rate $C=\dot m c_p$ . Before sizing area, confirm that supplied flow rates and terminal temperatures can satisfy that balance.

Arrangement controls the driving differences.

Counterflow pairs hot inlet with cold outlet at one end, and hot outlet with cold inlet at the other. Parallel flow uses different pairs. The logarithmic mean captures the changing local temperature difference. A correction factor may be needed for a specified multipass or crossflow arrangement.

Unknown outlet temperatures call for another route.

When outlet temperatures are unknown, effectiveness and NTU can relate duty to capacity rates and $UA$ . The maximum possible duty uses $C_{\min}$ , not the sum of the two capacity rates. Overall U and area must refer to the same side and include the stated resistance or fouling basis.

A decision map for this lesson

Pair temperatures at the same physical end

For counterflow, the cold-fluid inlet is opposite the hot-fluid inlet. The existing terminal diagram uses the worked-example temperatures.

A realistic double-pipe exchanger cutaway reveals separate inner-tube and annular flow passages and four fluid connections. — **Pair temperatures at the same physical end**Connect the equipment to the calculation

Counterflow exchanger: hot fluid enters at 180 degrees Fahrenheit on the left and exits at 140 on the right; cold fluid enters at 80 on the right and exits at 110 on the left. Terminal differences are 70 and 60. — **Pair temperatures at the same physical end**Connect the equipment to the calculation

Choose the governing relationship and catch the wrong route
Condition or question	Governing decision	Error to avoid
Both terminal temperatures are known	Pair the hot and cold temperatures at each physical end before finding LMTD.	Pairing both inlets merely because they are inlets.
Outlet temperatures are unknown	Use the appropriate effectiveness-NTU relation with the capacity rates and arrangement.	Inserting invented outlet temperatures into LMTD.
A supplied overall U is used	Match U and area to the same surface basis; include the stated correction factor.	Combining outside-area U with inside area without conversion.
Fouling resistance is specified	Add resistance on the consistent area basis and recalculate conductance.	Adding fouling resistance directly to U.

On a narrow screen, scroll across the table to compare the decision and the error.

Relationships and their limits

Counterflow terminal differences

\Delta T_1=T_{h,i}-T_{c,o};\quad\Delta T_2=T_{h,o}-T_{c,i}

For the stated counterflow arrangement; both driving differences must be physically consistent.

LMTD area method

\Delta T_{lm}=\frac{\Delta T_1-\Delta T_2}{\ln(\Delta T_1/\Delta T_2)};\qquad \dot Q=UAF\Delta T_{lm}

For equal terminal differences, use their common value. Apply the appropriate supplied correction factor F.

Effectiveness and NTU

\varepsilon=\frac{\dot Q}{C_{min}(T_{h,i}-T_{c,i})};\qquad NTU=UA/C_{min}

The relation connecting effectiveness to NTU depends on arrangement and capacity-rate ratio.

Symbols and unit basis
Symbol	Meaning	Units or basis
$U$	Overall coefficient on stated area basis	Btu/(h·ft²·°F)
$A$	Heat-transfer area on matching basis	ft²
$F$	Arrangement correction factor	dimensionless
$C$	Stream heat-capacity rate	Btu/(h·°F)
$\varepsilon$	Heat exchanger effectiveness	dimensionless
$\Delta T_{lm}$	Logarithmic mean driving difference	°F difference
$C_{min}$	Smaller stream heat-capacity rate	Btu/(h·°F)
$NTU$	Number of transfer units	dimensionless
$\dot Q$	Heat-transfer rate	Btu/h
$T_{h,i},\ T_{h,o}$	Hot-stream inlet and outlet temperatures	°F
$T_{c,i},\ T_{c,o}$	Cold-stream inlet and outlet temperatures	°F

One decision, fully worked

Apply the method.

A counterflow exchanger has hot water entering at 180°F and leaving at 140°F; cold water enters at 80°F and leaves at 110°F. U = 100 Btu/(h·ft²·°F), area = 30 ft² on the matching basis and F = 1. Find the predicted duty for these terminal temperatures.

Pair temperatures at the same physical ends of the counterflow exchanger.
Compute the logarithmic mean of 70°F and 60°F.
Multiply by the matching UA and the stated correction factor.

\Delta T_1=180-110=70^{\circ}\mathrm F;\quad\Delta T_2=140-80=60^{\circ}\mathrm F

\Delta T_{lm}=\frac{70-60}{\ln(70/60)}=64.87^{\circ}\mathrm F

\dot Q=100(30)(1)(64.87)\approx194{,}600\ \mathrm{Btu/h}

Decision

The predicted duty is about 194,600 Btu/h. To realize the specified temperatures, stream capacity rates must also support this duty.

Independent check

The logarithmic mean lies between 60°F and 70°F. Required hot and cold capacity rates are about 4,865 and 6,487 Btu/(h·°F), giving the same duty over their 40°F and 30°F changes.

Specialist extension

State the overall-U area basis before adding fouling

An overall heat-transfer coefficient and its area must refer to the same surface. If all resistances and fouling factors are supplied on a common area basis, fouling adds resistance and reduces U. For a cylindrical wall or unequal areas, convert terms before adding them. For LMTD calculations, pair temperatures at the same physical exchanger end and apply the supplied arrangement correction factor.

A new applied case

Clean U is 100 Btu/(h·ft²·°F). A new fouling resistance is 0.002 h·ft²·°F/Btu on the same area basis.

1/U_{dirty}=1/100+0.002=0.012

U_{dirty}=83.333\ \mathrm{Btu/(h\,ft^2\,^{\circ}F)}

Decision. Dirty U is about 83.333, on the supplied surface basis.

Cross-check. The added positive resistance must reduce U. It cannot increase heat transfer at otherwise fixed conditions.

Unit basis. U is Btu/(h·ft²·°F); fouling resistance is h·ft²·°F/Btu on the same area basis.

NCEES PE Mechanical Reference Handbook v2.1; see the module reference path.

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Transfer the method

HVAC Equipment and Components · HV-LC-E3

Try a new decision.

Select one

A counterflow exchanger has hot-fluid temperatures 170°F inlet and 130°F outlet; cold-fluid temperatures are 70°F inlet and 110°F outlet. U = 80 Btu/(h·ft²·°F), A = 25 ft² on the matching basis and F = 1. The heat-transfer duty is most nearly:

AUndefined because LMTD is 0/0B200,000 Btu/hC120,000 Btu/hD80,000 Btu/h

A lesson check supports practice, not a readiness score. Your response stays in this page session and clears on reload.

Catch the common errors

Pairing hot inlet with cold inlet for a counterflow terminal difference.
Dividing by zero when both driving differences are equal instead of taking the limit.
Using an inside-area U with an outside area.

Handbook navigation drill

Find it, then explain it.

Search: Log Mean Temperature Difference; effectiveness

Find the LMTD and effectiveness sections. Sketch the two counterflow ends before writing the temperature differences.

Check your navigation

Section 5.5 supplies exchanger relationships. The correction factor and effectiveness relation depend on arrangement; they are not interchangeable constants.

Reference path: §5.5, pp. 312–314. These are printed handbook page numbers; section names and search terms remain the primary navigation method. See technical review status.

Deeper application · Decision Lab

Unknown outlet temperatures: choose the exchanger model

Choose a method before assuming an outlet temperature. Then test whether changing the flow arrangement preserves the result.

Choose the method, predict the changed-condition result and inspect the complete cross-check. Sign in before starting to keep your first response and later reflection in the separate reasoning notebook.

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Theory sources

NCEES PE Mechanical Reference Handbook v2.1: §5.5, pp. 312–314
Mechanical and Electrical Equipment for Buildings, 12th ed.: mechanical systems and energy principles. Supporting theory; not an examination reference.

Worked examples and lesson prose are original ReasonSpan^™ material. Obtain authorized publications separately; this preview does not reproduce the supplied books or handbook.