Heat exchanger design uses temperature differences, heat transfer area, and fluid properties to predict how much heat can move between hot and cold streams. The LMTD method is one of the main tools engineers use when inlet and outlet temperatures are known or can be estimated. This cheat sheet helps students organize the formulas needed to size a heat exchanger and compare flow arrangements.
It is especially useful for solving steady-state design problems in thermal systems.
Key Facts
- The heat duty for a fluid with no phase change is q = m_dot cp (T_out - T_in), using the sign convention that heat gained is positive for the cold fluid.
- For an ideal heat exchanger with no heat loss to the surroundings, q_hot = q_cold in magnitude.
- The basic LMTD design equation is q = U A F DeltaT_lm, where U is overall heat transfer coefficient, A is area, F is correction factor, and DeltaT_lm is log mean temperature difference.
- The log mean temperature difference is DeltaT_lm = (DeltaT_1 - DeltaT_2) / ln(DeltaT_1 / DeltaT_2), where DeltaT_1 and DeltaT_2 are the terminal temperature differences.
- For counterflow, DeltaT_1 = T_h,in - T_c,out and DeltaT_2 = T_h,out - T_c,in.
- For parallel flow, DeltaT_1 = T_h,in - T_c,in and DeltaT_2 = T_h,out - T_c,out.
- Overall thermal resistance can be written as 1 / U = 1 / h_i + R_wall + 1 / h_o + R_f,i + R_f,o for a simplified flat-wall model.
- The effectiveness-NTU method uses effectiveness = q / q_max, q_max = C_min (T_h,in - T_c,in), and NTU = U A / C_min.
Vocabulary
- Heat exchanger
- A device that transfers thermal energy between two fluids without requiring the fluids to mix.
- LMTD
- The log mean temperature difference, which represents the average driving temperature difference across a heat exchanger.
- Overall heat transfer coefficient
- A combined measure of convection, wall conduction, and fouling resistance that controls heat transfer rate per area per temperature difference.
- Fouling
- The buildup of deposits on heat transfer surfaces that increases thermal resistance and lowers heat exchanger performance.
- Correction factor
- A multiplier used with LMTD to account for non-ideal flow arrangements such as shell-and-tube or crossflow designs.
- Effectiveness
- The ratio of actual heat transfer to the maximum possible heat transfer for the same inlet temperatures.
Common Mistakes to Avoid
- Using arithmetic average temperature difference instead of LMTD, which is wrong because the temperature driving force changes nonlinearly along the heat exchanger.
- Mixing up counterflow and parallel-flow terminal differences, which gives the wrong DeltaT_1 and DeltaT_2 and can greatly change the required area.
- Forgetting the correction factor F, which overpredicts heat transfer when the exchanger is not a simple ideal parallel-flow or counterflow unit.
- Ignoring fouling resistance, which makes U too large and causes the calculated heat transfer area to be too small for real operation.
- Using inconsistent units for m_dot, cp, U, and area, which makes q = U A F DeltaT_lm numerically incorrect even if the formula is chosen correctly.
Practice Questions
- 1 A counterflow heat exchanger has T_h,in = 120 C, T_h,out = 70 C, T_c,in = 25 C, and T_c,out = 55 C. Calculate DeltaT_1, DeltaT_2, and DeltaT_lm.
- 2 A heat exchanger transfers 45,000 W with U = 300 W/m^2 C, F = 0.90, and DeltaT_lm = 50 C. What heat transfer area is required?
- 3 Water flows at 0.80 kg/s with cp = 4180 J/kg C and warms from 20 C to 38 C. Calculate the heat gained by the water.
- 4 Why does counterflow usually allow a smaller heat exchanger area than parallel flow for the same inlet temperatures and heat duty?
Understanding Heat Exchanger Design (LMTD Method)
The temperature driving force is not the same at every point along a heat exchanger. It is usually large near one end and smaller near the other. The log mean temperature difference gives one carefully weighted average for this changing temperature gap.
It is not an ordinary arithmetic average because heat transfer responds more strongly where the temperature gap is larger. Before using this value, label each stream clearly and match the two temperatures at the same physical end of the exchanger. A common error is pairing inlet temperatures automatically, even when the fluids move in opposite directions.
Both terminal temperature differences must stay positive for the usual LMTD calculation. If one becomes zero or negative, the proposed outlet temperatures may be impossible for that flow arrangement.
Flow direction changes what temperatures are achievable. In parallel flow, both fluids enter from the same end. Their temperatures move closer together quickly, so the cold outlet cannot normally rise above the hot outlet.
Counterflow sends the fluids in opposite directions. It keeps a useful temperature difference over more of the length. This often allows the cold stream to leave at a temperature higher than the hot stream outlet.
That result can seem wrong at first, but energy is still conserved because each location has the hot fluid warmer than the nearby cold fluid. Students should sketch a temperature versus length graph before selecting an equation. The graph makes terminal differences, temperature cross, and the advantage of counterflow much easier to see.
The overall heat transfer coefficient represents several barriers in series. Heat first moves from the hot fluid to the wall by convection. It conducts through the wall material.
It then convects from the wall into the cold fluid. A low convection coefficient on either side can control the whole process. Thick walls and materials with poor thermal conductivity add resistance.
Deposits such as scale, rust, oil, or biological growth add fouling resistance. This matters in boilers, car radiators, refrigeration systems, and factory equipment.
Fouling slowly reduces heat duty unless more area or a larger temperature difference is available. Real designs include a fouling allowance because a clean exchanger does not stay clean forever.
The correction factor is needed when the exchanger has a more complex path, such as a shell-and-tube unit with several tube passes or a crossflow radiator. Its value is below one because the actual temperature pattern is less favorable than ideal counterflow. Using a correction factor close to one without checking the chart can produce an undersized design.
The LMTD method works best when all inlet and outlet temperatures are known. If outlet temperatures are unknown, the effectiveness and number of transfer units method is often more useful. It starts by finding the maximum possible heat transfer from the stream with the smaller heat capacity rate.
Comparing the actual duty with that limit shows whether more area will help much or whether one fluid flow rate is the main restriction. Always check units, use consistent temperature differences, and confirm that the heat lost by one stream matches the heat gained by the other within reasonable measurement error.