---
title: "Enthalpy formulation and heat transfer equation"
subject: "Applied Computational Fluid Dynamics"
chapter: 2
key: "semester_7/applied_computational_fluid_dynamics:enthalpy-energy-equation-heat-transfer"
source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/enthalpy-energy-equation-heat-transfer
site: BCT Engineering
---

# Enthalpy formulation and heat transfer equation

Applied Computational Fluid Dynamics · Chapter 2: Governing Equations and Numerical Foundations

- Asked in: year not given, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/enthalpy-energy-equation-heat-transfer
- Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-2

## Question

Write the energy equation in terms of enthalpy for a viscous flow and explain its terms. Show how it simplifies to the heat transfer (temperature) equation for an incompressible fluid with constant properties.

## Short answer

The enthalpy equation is ρ Dh/Dt = Dp/Dt + ∇·(k∇T) + Φ + S_h, representing enthalpy change, pressure work, conduction, viscous dissipation, and heat source. For incompressible constant-property flow with negligible dissipation and pressure work, it simplifies to ρcₚ DT/Dt = k∇²T + q̇, a convection–diffusion temperature equation.

## Answer

The energy equation comes from the first law of thermodynamics applied to a fluid element: the rate of change of energy equals the heat added plus the work done on the element.

#### Enthalpy form
With specific enthalpy $h=e+p/\rho$, the transport equation is
$$
\rho\frac{Dh}{Dt}=\frac{Dp}{Dt}+\nabla\cdot(k\nabla T)+\Phi+S_h
$$
that is
$$
\rho\left(\frac{\partial h}{\partial t}+\vec V\cdot\nabla h\right)=\frac{\partial p}{\partial t}+\vec V\cdot\nabla p+\nabla\cdot(k\nabla T)+\Phi+S_h
$$

| Term | Meaning |
|---|---|
| $\rho\,Dh/Dt$ | Rate of change of enthalpy of a moving fluid element |
| $Dp/Dt$ | Work of pressure changes; important for compressible flow |
| $\nabla\cdot(k\nabla T)$ | Heat conduction (Fourier's law) |
| $\Phi$ | Viscous dissipation: mechanical work converted to heat |
| $S_h$ | Volumetric heat source (reaction, radiation, Joule heating) |

For an ideal gas $dh=c_p\,dT$. For an incompressible liquid $dh=c\,dT+dp/\rho$, and the $dp/\rho$ part cancels the $Dp/Dt$ term. Either way, the equation can be written in terms of temperature.

#### Simplification
For incompressible flow with constant $\rho,\ c_p,\ k$ and negligible viscous dissipation and pressure work (low speed):
$$
\rho c_p\left(\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}\right)=k\nabla^2T+\dot q
$$
or
$$
\frac{\partial T}{\partial t}+\vec V\cdot\nabla T=\alpha\nabla^2T+\frac{\dot q}{\rho c_p},\qquad \alpha=\frac{k}{\rho c_p}
$$
This is the **convection-diffusion equation** for temperature.

#### Special cases
- No flow ($\vec V=0$): $\partial T/\partial t=\alpha\nabla^2T$, the transient heat conduction equation.
- Steady, no flow, no source: $\nabla^2T=0$, the Laplace equation.
- Steady 1D conduction with source: $k\,d^2T/dx^2+\dot q=0$.

The ratio of convection to conduction is the Peclet number $Pe=\rho c_pVL/k=Re\,Pr$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The energy equation comes from:
   A. Newton's second law
   B. Conservation of mass
   C. The first law of thermodynamics for a fluid element
   D. The second law of thermodynamics
   Answer: C. Rate of change of energy equals heat added plus work done on the element. (Key point: Basis of energy equation)

2. The term Phi in the enthalpy equation represents:
   A. Heat conduction
   B. Pressure work
   C. A volumetric heat source
   D. Viscous dissipation: mechanical work converted to heat
   Answer: D. Phi is the conversion of mechanical energy into heat by viscous friction. (Key point: Viscous dissipation)

3. Which conditions simplify the equation to the temperature equation?
   A. Incompressible, constant properties, negligible dissipation and pressure work
   B. High-speed compressible flow
   C. Variable density
   D. Large viscous heating
   Answer: A. With dh = cp dT and negligible dissipation and Dp/Dt, the result is the convection-diffusion equation. (Key point: Simplification conditions)

4. Setting V = 0 in the temperature equation gives:
   A. The Laplace equation always
   B. The continuity equation
   C. dT/dt = alpha del^2 T, transient conduction
   D. The Euler equation
   Answer: C. Without flow the convective term vanishes, leaving transient heat conduction. (Key point: Special case)

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