Thermal contact resistance creates a temperature jump at an interface; thermal equilibrium assumes there is no temperature jump. In a pilot plant heat exchanger or reactor wall, two adjacent phases can transfer heat continuously while having different interface temperatures when the boundary has finite resistance. The equilibrium approximation sets the temperatures equal, (T_1=T_2), and is valid only when the interface resistance is negligible compared with the surrounding conduction and convection resistances.

Thermal equilibrium at an interface is a modeling assumption of temperature continuity, whereas thermal contact resistance is a physical barrier represented by a finite temperature difference for a given heat flux. Roughness, oxidation, trapped air, imperfect clamping, or an interstitial material can make that barrier important in real pilot plant equipment.

What the Two Concepts Mean

Thermal equilibrium at a boundary

For two adjacent materials assumed to be in local thermal equilibrium, the interface temperature is continuous:

[ T_1 = T_2 ]

Heat can still flow through the interface if the surrounding system has a temperature gradient. Equal interface temperatures do not mean that the entire heat exchanger or reactor is at equilibrium or that the heat-transfer rate is zero.

A true system-wide thermodynamic equilibrium would require no net heat-transfer driving force. In heat-transfer modeling, “interface thermal equilibrium” usually means only that the two sides share the same local temperature.

Thermal contact resistance

Thermal contact resistance exists when heat encounters an additional resistance at the boundary between two regions. The resulting temperature difference is related to heat flux by the surface conductance condition:

[ q_{n1}=h_s(T_1-T_2)=q_{n2} ]

Here, (h_s) is the thermal contact conductance, and its reciprocal is the area-specific contact resistance:

[ R_c''=\frac{1}{h_s} ]

The interface therefore has a temperature jump:

[ \Delta T_{\text{interface}}=T_1-T_2=\frac{q_n}{h_s} ]

The larger the heat flux or the smaller the contact conductance, the larger the temperature difference across the interface.

Why Real Pilot Plant Interfaces Deviate from Equilibrium

Solid-solid interfaces

Solid-solid contact is rarely complete at the microscopic scale. Machined or unpolished surfaces touch primarily at asperities, while small gaps between those asperities may contain air.

Because air conducts heat poorly compared with metals, these gaps create a localized thermal resistance. Oxide layers, contamination, surface damage, and insufficient mechanical clamping can increase the resistance further.

Examples include:

Clamped heat-exchanger plates
Flanged or bolted reactor-wall assemblies
Removable vessel jackets
Metal liners or inserts
Contact regions between a reactor wall and an external heating or cooling element

At such interfaces, the metal temperatures immediately on either side need not be equal even though the heat flux is continuous.

Solid-fluid interfaces

At a solid-fluid boundary, the conventional engineering model often uses temperature continuity at the wall and represents the dominant temperature difference through a fluid-film convection resistance:

[ R_{\text{conv}}=\frac{1}{hA} ]

The fluid bulk temperature can therefore differ substantially from the wall temperature without implying contact resistance. This difference is normally attributed to the thermal boundary layer and is included in the convection coefficient (h).

A separate solid-fluid interfacial resistance may become relevant when the physical interface is imperfect, such as with a gas gap, fouling layer, poorly wetted surface, coating, deposit, or unusual microscale effects. In that case, the temperature immediately inside the fluid-side interface and the temperature at the solid surface may also differ.

The key distinction

For a solid-solid boundary, engineers often ask whether the two solid surfaces can be treated as having one common temperature. Contact resistance determines whether that simplification is justified.

For a solid-fluid boundary, engineers usually distinguish between the fluid bulk-to-wall temperature difference, modeled by convection, and any additional wall-to-fluid interfacial temperature jump, modeled by an interfacial resistance.

How the Difference Appears in a Heat-Transfer Model

Equilibrium boundary condition

Under the idealized equilibrium assumption, the interface conditions are:

[ T_1=T_2 ]

and

[ q_{n1}=q_{n2} ]

The heat flux remains continuous, but no temperature drop is assigned to the interface itself. The adjacent materials are treated as thermally connected without a distinct boundary resistance.

This approximation is appropriate when the interface resistance is much smaller than the bulk conduction and convection resistances.

Finite-conductance boundary condition

With contact resistance, temperature continuity is replaced by:

[ q_n=h_s(T_1-T_2) ]

The normal heat flux remains continuous in steady state:

[ q_{n1}=q_{n2}=q_n ]

but the temperatures differ across the interface. This condition is the thermal equivalent of placing a resistor between two nodes in a resistance network.

Incorporating contact resistance into total resistance

For a tubular heat exchanger, a simplified resistance model may contain the hot-side convection resistance, metal-wall conduction resistance, and cold-side convection resistance:

[ R_{\text{total}}

\frac{1}{h_iA_i} + \frac{x_m}{k_mA_m} + \frac{1}{h_oA_o} ]

If a contact interface exists, its resistance must be added:

[ R_{\text{total}}

R_i+R_c+R_m+R_o ]

For an interface with area (A_c):

[ R_c=\frac{1}{h_sA_c} ]

The heat-transfer rate is then:

[ Q=\frac{\Delta T}{R_{\text{total}}} ]

or, in overall-coefficient form:

[ Q=UA\Delta T ]

Adding (R_c) lowers the overall heat-transfer coefficient (U), assuming the other resistances and reference area remain consistently defined.

Why the Distinction Matters in Pilot Plants

Predicting actual temperature profiles

If contact resistance is omitted, the model may predict a single temperature at an interface where the equipment actually has a measurable temperature drop. This can produce inaccurate metal temperatures, fluid outlet temperatures, or reactor-wall temperature profiles.

The error is especially relevant when the interface resistance is comparable to the metal-wall or fluid-film resistances.

Interpreting temperature measurements

A sensor attached to one side of a wall may not measure the temperature on the other side. In a reactor or jacketed vessel, a measured external wall temperature may therefore differ from the internal wall temperature because of wall conduction and contact resistance.

This matters when evaluating reaction-temperature control, thermal degradation, hot spots, or cooling-system performance.

Estimating overall heat-transfer coefficients

Pilot plant data are often used to estimate an overall coefficient (U). If an unmodeled contact resistance exists, it may be incorrectly attributed to fouling, poor convection, low metal conductivity, or an inaccurate area estimate.

Separating the individual resistances improves scale-up analysis and helps identify the actual performance limitation.

Understanding the Trade-offs

When assuming equilibrium is reasonable

The equilibrium assumption simplifies the model and avoids introducing uncertain interface data. It is generally defensible when surfaces are clean, flat, tightly clamped, well wetted, and made with a highly conductive interface or when contact resistance is demonstrably small.

It is also useful when the interface resistance is insignificant relative to the fluid-film and wall-conduction resistances.

When the assumption becomes risky

The assumption becomes questionable for rough, oxidized, contaminated, or poorly fitted surfaces. It is also risky when air-filled gaps, coatings, deposits, or inadequate clamping are present.

A small physical gap can create a large thermal resistance because the gap material may conduct heat far less effectively than the adjoining metal.

The uncertainty in (h_s)

Thermal contact conductance is not usually a universal material constant. It depends on surface roughness, contact pressure, cleanliness, oxidation, temperature, interface material, and the actual contact area.

Using a generic value without checking the equipment configuration can create false precision. Where the interface is important, (h_s) should be obtained from equipment data, validated correlations, or calibration against measured temperatures and heat duties.

Avoiding double counting

A solid-fluid model should not count the same temperature drop twice. The bulk-fluid-to-wall difference belongs in the convection resistance, while a distinct physical interfacial layer or imperfect boundary belongs in the contact or interfacial resistance.

Likewise, a contact resistance should be added only where two regions are physically separated by a meaningful imperfect contact; it should not be inserted merely because two materials have different conductivities.

Making the Right Choice for Your Goal

Use the simplest boundary condition that remains consistent with the physical construction and the accuracy required.

If your primary focus is preliminary heat-duty estimation: Assume interface thermal equilibrium when the surfaces are clean, well fitted, tightly clamped, and the interface resistance is negligible relative to convection and wall conduction.
If your primary focus is accurate wall or surface temperatures: Include a finite contact conductance whenever roughness, oxidation, air gaps, coatings, deposits, or uncertain mechanical contact may cause a meaningful temperature jump.
If your primary focus is pilot-to-scale-up comparison: Add contact resistance to the total resistance network and avoid attributing its effect incorrectly to convection, fouling, or material conductivity.
If your primary focus is solid-fluid heat transfer: Model the bulk-fluid-to-wall difference with the appropriate convection coefficient, and add a separate interfacial resistance only when the physical boundary justifies it.

The practical rule is simple: thermal equilibrium makes interface temperatures equal by assumption, while thermal contact resistance preserves heat-flux continuity but permits a real temperature difference.

Summary Table:

Aspect	Thermal Equilibrium	Thermal Contact Resistance
Interface temperature	Assumes equal temperatures: `T1 = T2`	Allows a temperature jump: `T1 - T2 = qn/hs`
Heat flux	Continuous across the interface	Continuous across the interface in steady state
Model condition	Temperature continuity	Finite-conductance boundary: `qn = hs(T1 - T2)`
Main causes	Clean, flat, well-fitted, tightly clamped surfaces	Roughness, oxidation, air gaps, coatings, deposits, or poor clamping
Solid-fluid application	Bulk-fluid-to-wall difference is modeled with convection	Add interfacial resistance only when a separate physical barrier exists
Effect on heat transfer	No additional interface resistance	Adds `Rc = 1/(hsAc)` and lowers the overall heat-transfer coefficient `U`
Best use	Preliminary estimates when resistance is negligible	Accurate wall temperatures, pilot-scale analysis, and equipment diagnosis

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