For viscous fluid flows, the energy balance is formulated by separating total energy into thermal and mechanical contributions. The total-energy equation tracks internal, kinetic, and potential energy, while accounting for heat conduction and work performed by pressure and viscous stresses. Subtracting the mechanical-energy balance exposes the thermal effects of compression or expansion and the irreversible conversion of mechanical energy into heat through viscous dissipation.

The governing formulation combines a total-energy balance with a mechanical-energy balance. For viscous fluids, the internal-energy equation must include heat conduction, pressure-work effects, and the viscous dissipation term (\boldsymbol{\tau}:\nabla\mathbf{v}), which can be important in high-viscosity or high-shear operations.

Start With the Total-Energy Balance

Energy of a Fluid Particle

The substantial total energy per unit mass is

[ e_t = u + \frac{v^2}{2} + gz ]

where (u) is internal energy, (v^2/2) is kinetic energy, and (gz) is potential energy.

For a fluid particle, the first-law balance can be written as

[ \rho \frac{D e_t}{D t}

-\nabla\cdot\mathbf{q} + \nabla\cdot(\boldsymbol{\sigma}\cdot\mathbf{v}) + \rho \mathbf{b}\cdot\mathbf{v} + \dot{q}_v ]

where (\rho) is density, (\mathbf{q}) is conductive heat flux, (\boldsymbol{\sigma}) is the total stress tensor, (\mathbf{b}) is body force per unit mass, and (\dot{q}_v) represents any volumetric heat source.

Heat Conduction

Fourier’s law defines the conductive heat flux:

[ \mathbf{q}=-k\nabla T ]

Therefore, the conductive contribution to the energy balance is

[ -\nabla\cdot\mathbf{q}

\nabla\cdot(k\nabla T) ]

For constant thermal conductivity, this becomes (k\nabla^2T).

At a stationary solid wall, the fluid velocity at the wall is zero under the no-slip condition. Heat transfer across the boundary is consequently represented by the wall-normal conductive flux:

[ q_w=-k\left(\frac{\partial T}{\partial n}\right)_w ]

The wall temperature gradient determines the local heat-transfer rate used in heat exchangers, evaporators, cooling coils, and reactor jackets.

Stress Work

The stress tensor is commonly decomposed as

[ \boldsymbol{\sigma}=-P\mathbf{I}+\boldsymbol{\tau} ]

where (P) is pressure and (\boldsymbol{\tau}) is the deviatoric, or viscous, stress tensor.

The stress-work term includes both pressure work and viscous-stress work. This distinction is essential because pressure work primarily represents reversible mechanical energy transfer, whereas viscous stress can irreversibly generate thermal energy.

Derive the Thermal Energy Balance

Internal Energy Form

After separating kinetic and potential energy from the total-energy equation and using the momentum balance, the internal-energy equation is

[ \rho\frac{D u}{D t}

-\nabla\cdot\mathbf{q}

P\nabla\cdot\mathbf{v} + \boldsymbol{\tau}:\nabla\mathbf{v} + \dot{q}_v ]

Here, (\mathbf{v}) is the velocity field and the colon denotes the double contraction of two tensors.

The terms have direct physical meanings:

• (-\nabla\cdot\mathbf{q}): net conductive heat entering the fluid element.
• (-P\nabla\cdot\mathbf{v}): reversible pressure-volume work.
• (\boldsymbol{\tau}:\nabla\mathbf{v}): irreversible viscous conversion of mechanical energy into internal energy.
• (\dot{q}_v): externally specified volumetric heat generation.

Temperature-Pressure Form

For a thermodynamically characterized fluid, the internal-energy differential can be expressed in terms of temperature and pressure. A useful form of the thermal balance is

[ \rho c_p\frac{D T}{D t}

-\nabla\cdot\mathbf{q} + \left[ 1- \frac{T}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_P \right] \frac{D P}{D t} + \boldsymbol{\tau}:\nabla\mathbf{v} + \dot{q}_v ]

The exact pressure coefficient depends on the thermodynamic variables and equation of state used. The important physical contributions are the pressure-induced thermal effect and viscous dissipation.

For constant (k), the conductive term may be written as

[ -\nabla\cdot\mathbf{q}=k\nabla^2T ]

Compressibility and Pressure Effects

The term involving (DP/Dt) describes reversible heating or cooling associated with compression and expansion. It matters most when density changes appreciably, such as in gases, compressible liquids, rapid pressure transients, or strongly pressurized equipment.

For an incompressible liquid with nearly constant properties, this pressure-temperature contribution is often negligible in the thermal equation. Pressure can still be important in the separate mechanical-energy balance.

Account for Viscous Dissipation

General Viscous-Heating Term

The term

[ \Phi_v=\boldsymbol{\tau}:\nabla\mathbf{v} ]

is the rate at which viscous mechanical energy is converted into heat per unit volume.

It is nonnegative for a physically admissible Newtonian fluid and represents irreversible internal heating caused by velocity gradients.

Newtonian Fluids

For a Newtonian fluid,

[ \boldsymbol{\tau}

\mu\left[ \nabla\mathbf{v} + (\nabla\mathbf{v})^T \right] + \lambda(\nabla\cdot\mathbf{v})\mathbf{I} ]

where (\mu) is dynamic viscosity and (\lambda) is the second viscosity coefficient.

The dissipation function is therefore

[ \Phi_v=\boldsymbol{\tau}:\nabla\mathbf{v} ]

For incompressible flow with constant viscosity, it is commonly expressed using the rate-of-deformation tensor (\mathbf{D}):

[ \Phi_v=2\mu,\mathbf{D}:\mathbf{D} ]

Why Viscosity Matters in Unit Operations

In low-viscosity, low-shear flows, viscous heating may be small compared with heat exchanged through the equipment wall. In polymers, heavy oils, concentrated biological fluids, and intensive mixing operations, it can become a major internal heat source.

The consequence is that the fluid temperature may rise even when the equipment is externally cooled. Omitting (\Phi_v) in such systems can lead to incorrect outlet temperatures, heat-transfer coefficients, residence-time predictions, and cooling-duty requirements.

Formulate the Mechanical-Energy Balance

Mechanical Energy per Unit Mass

The mechanical-energy content of a flowing fluid is commonly represented by pressure, kinetic, and potential terms:

[ \frac{P}{\rho}+\frac{v^2}{2}+gz ]

For steady flow through a unit operation, a practical mechanical-energy balance per unit mass is

[ \frac{P_1}{\rho} + \frac{v_1^2}{2} + gz_1 + w_s

\frac{P_2}{\rho} + \frac{v_2^2}{2} + gz_2 + F ]

where (w_s) is shaft work supplied to the fluid and (F) is the mechanical-energy loss per unit mass caused by friction and other irreversible effects.

Rearranging gives

[ w_s

\frac{P_2-P_1}{\rho} + \frac{v_2^2-v_1^2}{2} + g(z_2-z_1) + F ]

The sign of (w_s) depends on the convention used. The equation above defines positive (w_s) as work supplied to the fluid.

Differential Mechanical Balance

A differential form can be derived from the momentum equation by projecting the stress work onto the velocity field. It accounts for pressure-gradient work, body-force work, kinetic-energy changes, shaft work, and viscous losses.

In practical pilot-plant calculations, the integrated form is usually more useful because it directly relates pump or compressor work to pressure rise, velocity change, elevation change, and frictional loss.

Mechanical Balance for Incompressible Flow

For constant-density liquid flow, the commonly used expression is

[ \frac{\Delta P}{\rho} + \frac{\Delta v^2}{2} + g\Delta z + F

w_s ]

Multiplying by mass flow rate gives the shaft-power requirement:

[ \dot W_s

\dot m \left[ \frac{\Delta P}{\rho} + \frac{\Delta v^2}{2} + g\Delta z + F \right] ]

This relation supports pump selection and prediction of pressure losses through pipes, fittings, heat exchangers, filters, mixers, and other modules.

Connect Mechanical Losses to Thermal Generation

Friction Is Not Automatically a Wall Heat Flux

The friction term (F) represents mechanical-energy degradation. It should not automatically be inserted into the thermal balance as heat transferred through the wall.

Part of the lost mechanical energy is converted locally into internal energy through viscous dissipation. The remainder may be transported by the fluid and ultimately transferred to equipment surfaces or surroundings.

Local Versus Overall Balances

The differential thermal equation uses the local generation term

[ \boldsymbol{\tau}:\nabla\mathbf{v} ]

An overall mechanical balance uses the integrated loss (F). These quantities are related through spatial integration, but they are not interchangeable without considering the flow field and control volume.

This distinction prevents double-counting frictional heating when both thermal and mechanical balances are applied to the same unit operation.

Heat and Work Are Different Transfer Modes

Heat crosses a system boundary because of a temperature difference. Work crosses the boundary through mechanical, electrical, pressure, or shaft forces.

A pump or agitator supplies work to the process, while a cooling jacket removes heat. The fact that mechanical work may eventually appear as thermal energy does not make the original shaft-work transfer a heat transfer.

Understanding the Trade-offs

Incompressible Approximations

For many liquid unit operations, density, viscosity, and thermal conductivity are treated as constant. This simplifies the thermal balance to

[ \rho c_p\frac{D T}{D t}

k\nabla^2T + \boldsymbol{\tau}:\nabla\mathbf{v} + \dot{q}_v ]

This approximation is appropriate only when pressure-induced density and property changes are insignificant over the operating range.

Neglecting Viscous Dissipation

Dropping the viscous term reduces the equation to a conventional convection-conduction balance. That is often reasonable for low-viscosity fluids, but it can be unsafe for narrow passages, high rotational speeds, intense mixing, or strongly shear-thinning materials.

The relevant comparison is between viscous heat generation and wall heat removal, not viscosity alone.

Steady-State Assumptions

At steady state, local accumulation may vanish, but convective transport, conduction, pressure effects, viscous dissipation, and heat exchange can remain significant.

Steady state does not mean that the thermal balance is negligible. It means that the temperature field no longer changes with time at a fixed location.

Bernoulli-Type Simplifications

If viscous losses, shaft work, heat effects, and compressibility are negligible, the mechanical balance reduces to a Bernoulli-type relation. This is useful for idealized flow, but it is generally insufficient for sizing real pumps or analyzing high-viscosity equipment.

How to Apply This to Your Project

The correct balance depends on whether the objective is to predict temperature, calculate equipment work, or evaluate both effects together.

• If your primary focus is temperature prediction: Use the internal-energy or temperature equation with conduction, convection, pressure-induced thermal effects where relevant, viscous dissipation, and volumetric heat generation.
• If your primary focus is pump or compressor sizing: Use the mechanical-energy balance with pressure change, kinetic-energy change, elevation change, friction loss, and shaft work.
• If your primary focus is high-viscosity or high-shear operation: Retain (\boldsymbol{\tau}:\nabla\mathbf{v}) and evaluate whether viscous heating materially changes the temperature field.
• If your primary focus is compressible or rapidly pressurized flow: Retain the (DP/Dt) thermodynamic term and use a consistent equation of state and property model.
• If your primary focus is an overall unit-operation duty: Combine the local thermal balance with the integrated mechanical balance while avoiding double-counting frictional energy.

A reliable formulation keeps thermal transport, mechanical energy, and their irreversible conversion explicitly connected but physically distinct.

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