How Is Overall Thermal Resistance Calculated In A Tubular Heat Exchanger Pilot Plant? Essential Formula Guide

The overall thermal resistance in a tubular heat exchanger pilot plant is simply the sum of three thermal resistances arranged in series: the convection resistance of the hot fluid, the conduction resistance of the metal tube wall, and the convection resistance of the cold fluid. For a clean tube operating at steady state, this means (R_{\text{total}} = \frac{1}{h_i A_i} + R_{\text{wall}} + \frac{1}{h_o A_o}), where (h_i) and (h_o) are the film heat transfer coefficients and (A) represents the respective heat transfer areas. The overall heat transfer coefficient (U) is then the reciprocal of this total resistance multiplied by a reference area, enabling the classic design equation (Q = U A \Delta T).

The core takeaway is that thermal resistance is additive. By breaking the total resistance down into its convection and conduction components, you can pinpoint which part of the system governs the heat transfer rate – a skill that lies at the heart of pilot plant analysis and equipment design.

The Series Resistance Mindset

In any heat exchanger, heat energy must travel from the hot bulk fluid to the tube surface, through the solid metal, and out to the cold bulk fluid. Each step imposes a distinct thermal barrier. The series resistance model treats these barriers as independent obstacles stacked one after another, so the total impediment to heat flow is simply their sum.

Why Series Addition Works

Under steady-state conditions, the rate of heat flow (Q) is constant through every part of the system.
Because the same (Q) passes through each resistance, the overall temperature drop (\Delta T) divides among them proportionally.
The overall resistance is therefore the sum of the individual resistances – exactly like voltage drops in an electrical circuit.

The Three Essential Resistances for a Clean Tube

For a pilot-scale tubular exchanger with no fouling, you must account for:

Hot-side convection resistance ((R_i = 1/(h_i A_i)))
Wall conduction resistance ((R_m))
Cold-side convection resistance ((R_o = 1/(h_o A_o)))

Breaking Down Each Component

Convection Resistance of the Fluid Films

The convective film coefficient (h) captures how effectively a fluid transports heat to or from a solid surface.
Higher turbulence and thermal conductivity increase (h), reducing the film resistance.
Since the hot and cold films have different fluid properties, velocities, and geometries, their resistances are calculated individually using the appropriate inside area (A_i) or outside area (A_o).

Conduction Resistance of the Metal Wall

For a thin-walled tube, the wall resistance is often approximated using a plane-wall formula with the logarithmic mean area (A_{\text{lm}}):
(R_m = \frac{x_m}{k_m A_{\text{lm}}}), where (x_m) is the wall thickness and (k_m) the metal’s thermal conductivity.

For accurate cylindrical geometry, the resistance is derived from Fourier’s law in radial coordinates:
(R_m = \frac{\ln(d_o/d_i)}{2 \pi k_m L}).
This expression appears in the more rigorous overall coefficient formula based on the outer area (U_o): [ \frac{1}{U_o} = \frac{1}{h_o} + \frac{d_o \ln(d_o/d_i)}{2 k_w} + \frac{d_o}{d_i} \frac{1}{h_i} ] Pilot plant calculations often use this form to reveal how wall curvature influences the resistance when comparing thick-walled tubes to flat plates.

The Real-World Impact of Fouling

Adding Fouling Resistances

Real process fluids leave deposits – scale, corrosion, biological films – that add fouling resistances (R_{fi}) and (R_{fo}).
These resistances sit in series with the convective films, expanding the denominator of (U): [ \frac{1}{U_o} = \frac{1}{h_o} + R_{fo} + \frac{d_o \ln(d_o/d_i)}{2 k_w} + \frac{d_o}{d_i} R_{fi} + \frac{d_o}{d_i} \frac{1}{h_i} ] Fouling factors for common pilot plant fluids (e.g., 0.00025 m²·°C/W for cooling water) dramatically reduce (U) over time, making this a critical teaching point.

How Pilot Plants Demonstrate Fouling

Students measure a clean exchanger’s (U) and then track the decline as fouling accumulates.
By comparing the experimental (U) to the clean design value, they quantify the added resistance and learn why periodic cleaning is essential.

How to Calculate Overall Resistance from Pilot Plant Data

Step 1: Determine the Heat Duty (Q)

Measure the mass flow rate and inlet/outlet temperatures of one fluid.
Use (Q = \dot{m} c_p \Delta T) (assuming negligible heat loss) to obtain the actual heat transferred.

Step 2: Compute the Log Mean Temperature Difference (LMTD)

With all four terminal temperatures, calculate the logarithmic mean driving force (\Delta T_{\text{lm}}).
This accounts for the temperature variation along the tube length if the flow arrangement is countercurrent or co-current.

Step 3: Extract the Experimental (U)

Rearrange (Q = U A \Delta T_{\text{lm}}) to solve for the overall coefficient based on a known area:
(U_{\text{exp}} = \frac{Q}{A \Delta T_{\text{lm}}}).

Step 4: Calculate Individual Resistances (Wilson Plot)

By varying the flow rate on one side while keeping the other side constant, you can isolate that film coefficient.
The resulting Wilson plot allows separation of the wall and film resistances, turning the measured (U) into a powerful diagnostic tool.

Understanding the Trade-offs

Simplified vs. Cylindrical Wall Resistance

The plane-wall approximation using a mean area is easier to teach and perfectly adequate for thin-walled tubes.
However, for thick-walled tubes or precise research, the logarithmic resistance form is the only correct choice – the error from the approximation can exceed 5% for diameter ratios above 1.5.

Ignoring Fouling in Clean Experiments

Omitting fouling factors simplifies the calculation but renders the result an idealized value that will never match long-term plant performance.
Pilot plants often run with clean fluids initially to establish baseline film coefficients, then deliberately introduce fouling to study its effect.

Assumptions That Limit Accuracy

The series resistance model assumes steady state, constant fluid properties, and no axial conduction.
In dynamic start-ups or with highly viscous fluids, these assumptions break down, and more advanced numerical models become necessary.

Making the Right Choice for Your Goal

If your primary focus is educational demonstration: Use the simple three‑resistance sum with a plane‑wall approximation and a mean heat transfer area. This builds intuitive understanding of the series resistance concept without the mathematical complexity.

If your primary focus is design or research: Adopt the logarithmic wall resistance and explicitly include fouling factors. Always reference the final (U) to a specific surface area (usually outside) so the calculation is unambiguous.

If your primary focus is troubleshooting or performance monitoring: Calculate the experimental (U) from operational data and compare it to a clean‑condition model. The difference isolates the fouling resistance, pointing you directly to whether the exchanger needs chemical or mechanical cleaning.

Ultimately, the overall thermal resistance in a tubular heat exchanger pilot plant is a gateway concept: once you master the additive nature of these resistances, you can diagnose real equipment, validate heat transfer correlations, and make data‑driven design decisions with complete confidence.

Summary Table:

Resistance Component	Formula / Representation	Key Influencing Factors
Hot-Side Convection	$R_i = \frac{1}{h_i A_i}$	Fluid velocity, turbulence, inside heat transfer area
Wall Conduction	$R_m = \frac{\ln(d_o/d_i)}{2 \pi k_m L}$	Tube material conductivity ($k_m$), thickness, diameter ratio
Cold-Side Convection	$R_o = \frac{1}{h_o A_o}$	Fluid properties, flow rate, outside heat transfer area
Fouling (Optional)	$R_{fi}$ & $R_{fo}$	Scale accumulation, fluid purity, hours of operation

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