For most pilot-plant liquid operations and low-speed gas systems, viscous dissipation and pressure-work terms can be neglected when their estimated contribution is small compared with thermal storage, convection, and conduction. In practice, this usually means moderate velocities, modest velocity gradients, low-to-moderate fluid viscosity, and pressure changes that are small relative to the fluid’s thermal-energy change.

Core takeaway: Neglect these terms only after comparing their scales with the retained terms. Viscous dissipation becomes important mainly in high-shear, high-viscosity flows; pressure work becomes important mainly for compressible fluids undergoing substantial or rapid pressure changes.

Start with the Relevant Energy Equation

Temperature-based form

A useful local thermal-energy equation is

[ \rho c_p\frac{DT}{Dt}

k\nabla^2T + \beta T\frac{Dp}{Dt} + \Phi ]

where:

(\rho c_p DT/Dt) is the rate of thermal-energy storage and convection,
(k\nabla^2T) represents heat conduction,
(\beta T Dp/Dt) is the pressure-work or compression term,
(\Phi) is the viscous-dissipation function.

The terms may be omitted when each is small relative to the dominant terms in the specific unit operation.

The practical comparison

For a pilot plant, the correct question is not whether velocity or pressure variation exists. They almost always do.

The correct question is whether the associated energy contribution is large enough to affect the required accuracy of the model or experiment.

When Viscous Dissipation Can Be Neglected

Moderate velocity and velocity gradients

Viscous dissipation is generated by deformation of the fluid, not simply by bulk motion.

For a characteristic velocity (U) and length scale (L), its order of magnitude is commonly estimated as

[ \Phi \sim \mu\left(\frac{U}{L}\right)^2 ]

where (\mu) is dynamic viscosity.

It is generally negligible when (U/L) is moderate and the resulting (\Phi) is small compared with the thermal storage, convection, or conduction terms.

Low- and moderate-viscosity fluids

For water, dilute aqueous solutions, air, and similar fluids under ordinary pilot-plant conditions, viscous heating is normally very small.

This is typically true in:

Shell-and-tube heat exchangers,
Low-speed flow rigs,
Packed absorption columns,
Liquid-phase stirred tanks,
Standard natural-convection demonstrations.

The usual approximation is therefore

[ \rho c_p\frac{DT}{Dt} \approx k\nabla^2T + \beta T\frac{Dp}{Dt} ]

before determining whether the pressure term can also be removed.

Short flow paths and limited shear regions

Even when a process contains localized high-shear zones, viscous heating may remain unimportant if those zones are small or the residence time is short.

For example, a pump, valve, impeller tip, or narrow gap may produce local heating without materially changing the bulk temperature of the process stream.

A useful scale estimate

Compare viscous dissipation with thermal convection:

[ \frac{\Phi}{\rho c_p U\Delta T/L} \sim \frac{\mu U}{\rho c_p L\Delta T} ]

If this ratio is much less than one, viscous dissipation is unlikely to influence the bulk temperature field.

It can also be compared with conduction using a Brinkman-type scale:

[ \frac{\Phi}{k\Delta T/L^2} \sim \frac{\mu U^2}{k\Delta T} ]

The precise threshold depends on the required accuracy, but a value far below unity supports omission.

When Pressure Work Can Be Neglected

Thermodynamically incompressible liquids

For an idealized incompressible liquid,

[ \left(\frac{\partial \rho}{\partial T}\right)_p=0 ]

and the thermal expansion coefficient is effectively zero. Consequently,

[ \beta T\frac{Dp}{Dt}\approx 0 ]

This is why pressure-work effects are normally omitted for water-like liquids in ordinary pilot-scale piping, heat exchangers, and mixing tanks.

Strictly speaking, real liquids have finite compressibility and thermal expansion. The term is usually negligible because those properties are sufficiently small under ordinary operating conditions—not because pressure has no effect whatsoever.

Small pressure changes in gases

For gases, pressure work cannot automatically be discarded because gases are compressible.

A practical criterion is

[ \Delta P \ll \rho c_p\Delta T ]

where (\rho c_p\Delta T) represents the characteristic thermal-energy change per unit volume.

Equivalently, from the local equation, compare

[ \beta T\frac{\Delta P}{\tau} ]

with

[ \rho c_p\frac{\Delta T}{\tau} ]

so that

[ \frac{\beta T\Delta P}{\rho c_p\Delta T}\ll 1 ]

For an ideal gas, (\beta \approx 1/T), giving the simplified comparison

[ \Delta P \ll \rho c_p\Delta T ]

For ambient air, a temperature change of approximately (8\ \mathrm{K}) has an energy scale comparable to a pressure change of about (10^4\ \mathrm{Pa}), or (0.1) atm. This provides a useful engineering scale, not a universal cutoff.

Slow or modest compression

Pressure work is usually safe to omit when pressure changes are both modest and slow compared with the thermal response of the system.

This often applies to low-speed gas flow through pilot heat exchangers or process lines when the pressure drop is small relative to the absolute operating pressure and the temperature change is measurable.

Pressure changes that do not significantly alter the temperature field

A pressure drop may be important for pump power, flow rate, or mechanical design while remaining unimportant in the thermal equation.

These are separate questions: a pressure loss can dominate the hydraulic balance without producing a significant bulk temperature change through the local pressure-work term.

Pilot-Plant Conditions That Usually Justify Both Omissions

Standard liquid unit operations

Both terms can normally be neglected for water-like or moderately viscous liquids operating at ordinary temperatures and pressures in:

Heat exchangers,
Mixing tanks,
Packed columns,
Liquid reactors,
Routine piping and flow experiments.

The resulting equation is commonly written as

[ \rho c_p\frac{DT}{Dt}

k\nabla^2T + H_V ]

where (H_V) represents any externally specified volumetric heat source.

Low-speed gas operations

For air and other gases at approximately ambient conditions, both terms are often negligible when:

Flow velocities are well below sonic speed,
Pressure changes are small,
Temperature changes are not negligible compared with pressure-induced thermal effects,
The system is not undergoing rapid compression or expansion.

Ordinary natural-convection systems

In standard laboratory-scale natural-convection equipment, viscous dissipation is generally insignificant for common fluids and characteristic dimensions.

It may become relevant in unusual low-temperature gas systems, where the volumetric expansion coefficient can become large, but this is not representative of ordinary pilot-plant operation.

Understanding the Trade-offs

High-viscosity and high-shear fluids

Viscous dissipation should be retained for polymer melts, heavy oils, concentrated fermentation broths, and other highly viscous materials exposed to strong shear.

It is especially important in narrow channels, high-speed mixers, extruders, pumps, and lubrication-like flows.

Omitting (\Phi) in these cases can underpredict temperature rise and may lead to incorrect viscosity, heat-transfer, reaction-rate, or product-quality predictions.

Rapid compression and expansion

Pressure work should be retained in compressors, expanders, high-pressure gas systems, and rapidly changing gas flows.

It also deserves attention in cryogenic or very-low-temperature systems, where fluid properties and thermal expansion behavior can make pressure-induced thermal effects disproportionately important.

Nearly isothermal systems

A small pressure-work contribution may still matter if the ordinary thermal-energy change is even smaller.

Therefore, the criterion is not simply “small pressure drop.” Pressure work should be checked carefully in systems that are deliberately isothermal or have very small temperature gradients.

Large pressure changes in liquid systems

Although pressure work is commonly negligible for liquids, high-pressure processing, supercritical operations, and systems near phase boundaries may require a more complete thermodynamic treatment.

In such systems, compressibility, thermal expansion, enthalpy changes, and phase equilibrium effects should be evaluated rather than assumed away.

Avoid confusing local pressure work with enthalpy accounting

In a steady-flow control-volume energy balance, pressure effects may already be incorporated through enthalpy and flow-work terms.

The decision to omit (\beta T Dp/Dt) applies to the local temperature-field equation. It does not automatically justify removing pressure effects from pump, compressor, turbine, throttling, or phase-equilibrium calculations.

How to Apply This to Your Pilot Plant

Use a scale estimate before simplifying the model:

If your primary focus is routine liquid heat transfer or mixing: Treat the liquid as incompressible and omit pressure work; also omit viscous dissipation unless the fluid is highly viscous or the equipment produces strong shear.
If your primary focus is low-speed gas flow near ambient conditions: Omit both terms when (\Delta P/(\rho c_p\Delta T)\ll 1) and the velocity gradients are moderate.
If your primary focus is polymer, heavy-oil, or concentrated-broth processing: Estimate (\Phi) explicitly and retain viscous dissipation if its temperature rise is comparable with the measured or predicted thermal change.
If your primary focus is compression, expansion, cryogenic, or high-pressure operation: Retain pressure work and evaluate compressibility, property variation, and enthalpy effects directly.
If your primary focus is high-accuracy pilot-plant correlation: Perform an order-of-magnitude check for both terms, even when standard practice suggests neglecting them.

A simplified energy equation is reliable when it is supported by scale analysis rather than by habit.

Summary Table:

Operating condition	Viscous dissipation	Pressure work	Typical applications
Water-like or dilute liquids	Usually negligible	Usually negligible	Heat exchangers, tanks, columns, piping
Low-speed gases near ambient conditions	Usually negligible with moderate shear	Negligible when ΔP/(ρcₚΔT) ≪ 1	Air-flow rigs, gas heat exchangers
High-viscosity fluids or strong shear	Retain and estimate Φ	Usually case-dependent	Polymer melts, heavy oils, concentrated broths
Rapid compression or expansion	Case-dependent	Retain	Compressors, expanders, high-pressure gas systems
Cryogenic, supercritical, or near-phase-boundary systems	Evaluate properties carefully	Retain	Cryogenic and high-pressure operations
High-accuracy pilot-plant studies	Verify with scale analysis	Verify with scale analysis	Research and process correlation

Need to determine whether your pilot plant energy model can safely omit viscous dissipation or pressure work? Contact LABPARK for expert support in selecting and configuring Educational and Vocational Unit Operations Pilot Plants for chemical engineering, bioprocess and biotech, and environmental and water treatment. LABPARK helps universities, research institutes, and enterprises evaluate realistic operating conditions, compare energy-equation assumptions, and build reliable training, research, and process-development systems.