Heat transfer notes
Thermal conduction
Thermal conduction is the transfer of heat in internal energy by microscopic collisions of
particles and movement of electrons within a body.
The differential form of Fourier's law of thermal conduction shows that the local heat flux
density, 𝑞/𝐴, is equal to the product of thermal conductivity, k, and the negative local
temperature gradient, − ∂T/ ∂x. The heat flux density is the amount of energy that flows
through a unit area per unit time.
𝑞
𝐴
𝑘
∂T
∂x
Where 𝑞 is the heat transfer per unit time [W], A is the area [𝑚 ], k is the materials conductivity
[W/(mK)], T is the temperature [K], and x is the length parameter [m].
Thermal convection
Convective heat transfer, often referred to simply as convection, is the transfer of heat from
one place to another by the movement of fluids. Convection is usually the dominant form of
heat transfer in liquids and gases.
Newton’s law of cooling describes a basic relationship for heat transfer by convection:
𝑞
𝐴
ℎ T
T
Where 𝑞 is the heat transfer per unit time [W], A is the area [𝑚 ], h is the heat transfer
coefficient [W/(m2K)], 𝑇 is the temperature of the fluid [K], and 𝑇 is the temperature on the
object’s surface [K].
Instationary heat conduction equation
In three dimensions:
𝜕
𝜕𝑇
𝜕
𝜕𝑇
𝜕
𝜕𝑇
𝜕𝑇
𝜌𝑐
𝑘
𝑘
𝑘
𝑄
𝜕𝑦 𝜕𝑦
𝜕𝑧 𝜕𝑧
𝜕𝑡 𝜕𝑥 𝜕𝑥
Where 𝑄
is the heat generation within the material, 𝜌 is the density, and 𝑐 is the specific
heat.
In one dimension:
𝜌𝑐
𝜕𝑇
𝜕𝑡
𝜕
𝜕𝑇
𝑘
𝜕𝑥 𝜕𝑥
𝑄
Stationary heat conduction equation
No temperature change in time:
𝜕𝑇
0
𝜕𝑡
Heat conduction equation in one dimension:
𝜕𝑇
𝜕
𝑘
0
𝜕𝑥 𝜕𝑥
𝑄
Biot number (Bi)
The Biot number is a dimensionless quantity used in heat transfer calculations. It gives a simple
index of the ratio of the heat transfer resistances inside of a body and at the surface of a body.
This ratio determines whether or not the temperatures inside a body will vary significantly in
space, while the body heats or cools over time, from a thermal gradient applied to its surface.
ℎ𝐿
𝐵𝑖
𝑘
Where L is the characteristic length, given by:
𝑉
𝐿
𝐴
Where V is the volume and Ac is the area from which the object cools.
Lumped analysis
Assuming an object has a constant temperature at a given time, the lumped analysis gives the
temperature of the object as a function of time:
ℎ𝐴 𝑡
𝑇 𝑡
𝑇
𝑇
𝑇
𝑒𝑥𝑝
𝜌𝑐 𝑉
Where Tinf is the temperature in the cooling medium.
Notes are based on these sources:
https://en.wikipedia.org/wiki/Thermal_conduction
https://en.wikipedia.org/wiki/Convective_heat_transfer
https://en.wikipedia.org/wiki/Biot_number