The overall heat transfer resistance is the sum of three primary thermal barriers in series.
In a shell-and-tube heat exchanger pilot plant, the total resistance ( R = 1/(U A) ) is calculated by adding the hot‑fluid convection resistance, the metal wall conduction resistance, and the cold‑fluid convection resistance. Under steady‑state operation, researchers measure fluid inlet/outlet temperatures and flow rates to compute heat duty, then derive the overall heat transfer coefficient ( U ) from ( Q = U A \Delta T_{lm} ). This experimental value reveals the aggregate impedance to heat flow and serves as the foundation for diagnosing which sub‑resistance controls the overall performance.
The real power of a pilot‑plant experiment lies not in obtaining a single ( U ) value, but in disaggregating the total resistance into its convective and conductive contributors. By systematically varying flow rates, wall materials, and fluid properties, students and engineers can identify the dominant bottleneck — often the fluid‑side film — and learn how boundary‑layer dynamics dictate the entire exchanger’s efficiency.
Understanding the Resistance Network
The Series Resistance Model
Imagine heat as a current that must cross three “resistors” placed end to end: the hot fluid film, the tube wall, and the cold fluid film. The total resistance to heat flow is simply the sum of these individual resistances:
[ R = \frac{1}{h_i A_i} + \frac{x_m}{k_m A_m} + \frac{1}{h_o A_o} ]
Here, ( h_i ) and ( h_o ) are the convective heat transfer coefficients on the inside and outside surfaces, ( A_i ) and ( A_o ) are the respective heat transfer areas, ( x_m ) is the tube wall thickness, ( k_m ) is the wall’s thermal conductivity, and ( A_m ) is a representative area for conduction (often the logarithmic mean area for a cylindrical tube). This additive structure — firmly anchored in the primary reference’s description — explains why a large convective resistance on one side can dominate the entire exchange process, even if the tube wall is highly conductive.
The Role of the Overall Heat Transfer Coefficient
The overall heat transfer coefficient ( U ) packages these individual resistances into a single lumped parameter.
[ \frac{1}{U A} = R \quad \Rightarrow \quad Q = U A \Delta T ]
In practice, ( U ) is always tied to a specific area: if ( U_o ) is based on the outside tube area, then each resistance term is scaled accordingly. Supplementary pilot‑plant literature expands this series to include fouling resistances (( h_{od} ) and ( h_{id} )), which act as additional insulating layers that grow over time. The core experimental task is to measure ( U ) accurately and then, through systematic variation, back out the individual components that constitute it.
Experimental Determination on a Pilot Plant
Measuring Heat Duty ( Q )
The first step is to operate the unit under true steady‑state conditions. Students measure the mass flow rates (( m )) and the inlet/outlet temperatures of both streams. The heat balance equation gives the actual heat load:
[ Q_h = m_h c_{p,h} (T_{h,in} - T_{h,out}), \quad Q_c = m_c c_{p,c} (T_{c,out} - T_{c,in}) ]
The two values should agree within a small margin (typically <5%), confirming that heat losses to the surroundings are negligible. This cross‑check is a critical quality control moment in any pilot‑plant session.
Calculating the Driving Force: LMTD
Because the temperature difference between the hot and cold sides changes along the exchanger length, the correct driving force is the log mean temperature difference (LMTD):
[ \Delta T_{lm} = \frac{(T_{h,in} - T_{c,out}) - (T_{h,out} - T_{c,in})}{\ln\left( \frac{T_{h,in} - T_{c,out}}{T_{h,out} - T_{c,in}} \right)} ]
For co‑current flow the formula adjusts accordingly, but the LMTD always ensures a thermodynamically consistent driving force. The primary reference’s generic ( \Delta T ) is almost universally replaced by the LMTD in rigorous pilot‑plant analysis.
Solving for ( U ) and Total Resistance
With ( Q ) and ( \Delta T_{lm} ) known and the heat transfer area ( A ) fixed by the equipment geometry, the overall coefficient is directly computed:
[ U = \frac{Q}{A \Delta T_{lm}} ]
The total resistance then follows as ( R = 1/(U A) ). At this stage, the student has a single experimental data point: the lumped ( U ). The real analysis begins when this value is compared under different operating conditions.
Decomposing the Overall Resistance
Isolating Convective Coefficients with the Wilson Plot
The Wilson plot technique is the gold standard in educational pilot plants for separating the inside and outside film coefficients. The method exploits the fact that the overall resistance changes predictably with fluid velocity. For a reactor or tube‑side measurement, the user plots ( 1/U ) against ( v^{-0.8} ) (or ( n^{-2/3} ) for stirred vessels). The y‑intercept extracts the sum of the stationary resistances (wall + outside film + fouling), while the slope yields the fluid‑side coefficient. This approach allows students to verify the classic Dittus–Boelter type correlations right on the pilot‑plant data.
Accounting for Fouling in Long‑Duration Runs
Freshly cleaned exchangers exhibit a “clean” ( U ). As experiments proceed, mineral deposits or product buildup add an extra series resistance — the fouling factor. By logging ( U ) periodically over several hours, students see a gradual decline in the overall coefficient. The difference between initial and final ( U ) values directly quantifies the fouling resistance. This teaches the practical lesson that pilot‑plant data must always be time‑stamped and that industrial design always includes a fouling allowance.
The Influence of Flow Configuration
Switching between co‑current and counter‑current flow does not alter the individual resistances themselves, but it changes the LMTD for a given set of terminal temperatures. Counter‑current flow yields a larger ( \Delta T_{lm} ), and thus a higher ( Q ) for the same ( U ). By running both configurations on the same pilot plant, students learn that maximizing thermal driving force is as important as minimizing the ( 1/h ) terms.
Understanding the Trade‑offs
The Assumption of Steady State
All resistance analysis assumes steady‑state heat transfer. In a pilot plant, achieving true thermal equilibrium can take 20–40 minutes. Rushing the measurements while temperatures are still drifting leads to an apparent ( U ) that does not reflect the real exchanger performance. The compromise is patience: the trade‑off is between lab session time and measurement accuracy.
Accuracy of Temperature Measurement
Small errors in thermocouple readings propagate dramatically into the LMTD, especially when the hot‑ and cold‑end approach temperatures are close. A 0.5 °C error can shift the calculated ( U ) by 10–15%. This forces a trade‑off: operating at large temperature differences improves precision but may not represent the mild conditions of an energy‑efficient industrial design.
The Hidden Cost of Neglecting Fouling
When students determine ( U ) from short‑run data and ignore fouling, they get an optimistic coefficient that will never be sustained in a real process. While skipping the fouling factor shortens the experiment, it misleads the design. The responsible approach is to acknowledge that the measured ( U ) is an instantaneous value and to emphasize that long‑duration runs are essential for reliability.
Making the Right Choice for Your Goal
- If your primary focus is understanding heat transfer fundamentals: Concentrate on the Wilson plot. Vary flow rates systematically and let the data teach you how ( h ) scales with Reynolds number.
- If your primary focus is designing an industrial heat exchanger: Run extended experiments with process‑representative fluids and record the ( U ) decay over time. Use the lowest sustained ( U ) to derive a realistic fouling factor.
- If your primary focus is optimizing energy recovery in an existing pilot plant: Compare co‑current and counter‑current configurations at identical flow rates. The counter‑current arrangement almost always delivers a higher LMTD for the same area, maximizing heat recovery without changing the hardware.
- If your primary focus is validating simulation models: Measure the wall temperature in addition to the fluid temperatures. This extra data point allows you to split the total resistance and check whether your CFD model predicts the correct film coefficients on each side.
A pilot‑plant heat exchanger is more than a piece of steel; it is a living example of series resistances that you can learn to measure, dissect, and ultimately control.
Summary Table:
| Resistance Type | Formula | Controlling Factors |
|---|---|---|
| Inside Convective Film | $1/(h_i A_i)$ | Fluid velocity, viscosity, tube geometry |
| Tube Wall Conduction | $x_m/(k_m A_m)$ | Wall thickness, material thermal conductivity |
| Outside Convective Film | $1/(h_o A_o)$ | Flow rate, shell-side geometry, baffles |
| Fouling Layer (Time-dependent) | $1/(h_{d} A)$ | Fluid cleanliness, temperature, runtime |
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