---
title: "General heat conduction equation in Cartesian coordinates"
subject: "Heat Transfer"
chapter: 2
key: "semester_5/heat_transfer:general-heat-conduction-equation-cartesian"
source: https://www.bctengineer.com/q/semester_5/heat_transfer/general-heat-conduction-equation-cartesian
site: BCT Engineering
---

# General heat conduction equation in Cartesian coordinates

Heat Transfer · Chapter 2: Conduction Heat Transfer

- Asked in: year not given, 8 marks
- Video: [General heat conduction equation for Cartesian co-ordinate](https://www.youtube.com/watch?v=BfYG3R3eo0c) — TECHNICAL CLASSES
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/heat_transfer/general-heat-conduction-equation-cartesian
- Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-2

## Question

Derive the general three-dimensional heat conduction equation in Cartesian coordinates for an isotropic solid with internal heat generation. Write its reduced forms for (a) steady state with no generation (Laplace), (b) steady state with generation (Poisson), and (c) unsteady state with no generation (Fourier). State the cylindrical-coordinate form.

## Short answer

General equation: ∇²T + q̇/k = (1/α) ∂T/∂t. Reduced forms: Laplace ∇²T = 0, Poisson ∇²T + q̇/k = 0, Fourier ∇²T = (1/α) ∂T/∂t. Cylindrical form: (1/r)∂/∂r(r ∂T/∂r) + (1/r²)∂²T/∂φ² + ∂²T/∂z² + q̇/k = (1/α) ∂T/∂t.

## Answer

#### Derivation
Take a small element $dx\,dy\,dz$ in a solid of density $\rho$, specific heat $c$ and constant conductivity $k$. Let $\dot q_g$ be the heat generated per unit volume (W/m$^3$). Apply the energy balance:

(heat in) + (heat generated) = (heat out) + (increase of internal energy).

Heat conducted in the $x$ direction at face $x$ (Fourier's law):

$$
Q_x = -k\,dy\,dz\,\frac{\partial T}{\partial x}
$$

Heat leaving at $x+dx$ (Taylor series):

$$
Q_{x+dx} = Q_x + \frac{\partial Q_x}{\partial x}dx = Q_x - k\,dy\,dz\,\frac{\partial^2 T}{\partial x^2}dx
$$

Net heat gained in $x$ direction is $k\,\dfrac{\partial^2 T}{\partial x^2}\,dx\,dy\,dz$. Similarly for $y$ and $z$. Adding generation $\dot q_g\,dx\,dy\,dz$ and equating to the rate of increase of stored energy $\rho c\,\dfrac{\partial T}{\partial t}\,dx\,dy\,dz$:

$$
k\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right)+\dot q_g = \rho c\frac{\partial T}{\partial t}
$$

Dividing by $k$ and defining thermal diffusivity $\alpha = k/\rho c$ (m$^2$/s):

$$
\nabla^2 T + \frac{\dot q_g}{k} = \frac{1}{\alpha}\frac{\partial T}{\partial t}
$$

#### Reduced forms
| Condition | Equation |
|---|---|
| Steady, no generation (Laplace) | $\nabla^2 T = 0$ |
| Steady, with generation (Poisson) | $\nabla^2 T + \dot q_g/k = 0$ |
| Unsteady, no generation (Fourier) | $\nabla^2 T = \dfrac{1}{\alpha}\dfrac{\partial T}{\partial t}$ |

#### Cylindrical coordinates $(r,\phi,z)$

$$
\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 T}{\partial \phi^2}+\frac{\partial^2 T}{\partial z^2}+\frac{\dot q_g}{k}=\frac{1}{\alpha}\frac{\partial T}{\partial t}
$$

For one-dimensional radial steady flow without generation this becomes $\dfrac{d}{dr}\left(r\dfrac{dT}{dr}\right)=0$.

## Check yourself

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

1. What energy balance is applied to the element?
   A. Heat in + generated = heat out + increase of internal energy
   B. Heat in = heat out only
   C. Heat in - generated = stored energy
   D. Heat out = generated + stored
   Answer: A. Net conduction in plus generation equals the rate of stored energy increase. (Key point: Energy balance)

2. How is thermal diffusivity defined?
   A. alpha = rho c/k
   B. alpha = k/(rho c)
   C. alpha = k rho/c
   D. alpha = c/(k rho)
   Answer: B. Dividing by k introduces alpha = k/(rho c) in m^2/s. (Key point: Thermal diffusivity)

3. Which is the Poisson equation?
   A. Laplacian T = 0
   B. Laplacian T = (1/alpha) dT/dt
   C. Laplacian T + qg/k = 0
   D. Laplacian T + qg/k = (1/alpha) dT/dt
   Answer: C. Poisson is steady state with generation: the time term drops and qg stays. (Key point: Reduced forms)

4. For 1-D steady radial flow without generation in a cylinder, which equation holds?
   A. d/dr(dT/dr) = 0
   B. d/dr(r^2 dT/dr) = 0
   C. r d^2T/dr^2 = qg/k
   D. d/dr(r dT/dr) = 0
   Answer: D. The cylindrical form reduces to d/dr(r dT/dr) = 0; r^2 is for a sphere. (Key point: Radial cylinder)

## Related in Heat Transfer

- [Plane wall with internal heat generation](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Aplane-wall-with-heat-generation) — Heat Transfer, chapter 2
- [Fourier law and thermal conductivity](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Afourier-law-thermal-conductivity) — Heat Transfer, chapter 2
- [Modes of heat transfer with rate equations](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Amodes-of-heat-transfer) — Heat Transfer, chapter 1
- [Hollow cylinder conduction and critical radius](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Acritical-radius-insulation-cylinder) — Heat Transfer, chapter 2
- [Conduction through hollow sphere](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Ahollow-sphere-heat-flow) — Heat Transfer, chapter 2
- [Dimensional analysis of forced convection](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Abuckingham-pi-forced-convection) — Heat Transfer, chapter 3

## Related in other subjects

- [Steady vs unsteady heat transfer; three-layer cylinder](https://www.bctengineer.com/question.md?key=semester_2%2Ffundamental_of_thermodynamics_and_heat_transfer%3Asteady-unsteady-composite-cylinder-three-layers) — Fundamental of Thermodynamics & Heat Transfer, chapter 7
- [Variational approach for one-dimensional heat conduction](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Avariational-approach-heat-conduction) — Finite Element Method, chapter 5
- [Continuity equation in Cartesian coordinates](https://www.bctengineer.com/question.md?key=semester_4%2Ffluid_mechanics_mechanical%3Acontinuity-equation-cartesian-derivation) — Fluid Mechanics, chapter 4
- [Derive one-dimensional heat equation](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Aheat-eq-derive) — Applied Mathematics, chapter 3, asked 5×