Heat Transfer¶

Definitions¶

System: what we want to study
Surroundings: external to system
Boundary: region between system and surroundings where interactions happen
Property: macroscopic characteristic of system
State: condition of system described by properties
Process: transformation from one state to another
Steady state: when a certain property does not change with time
Extensive property: overall value is sum of its parts (e.g. mass, volume, energy)
Intensive property: values are independent of size, extent (e.g. pressure, temperature)

Thermodynamic Systems¶

Isolated system
- No exchange of energy
- No exchange of matter
Closed system
- Exchange of energy
- No exchange of matter
Open system/Control volume
- Exchange of energy
- Exchange of matter

Work Done and Heat Transfer¶

Work Done, W¶

W > 0: Work done BY system
W < 0: Work done ON system

Warning

Work done is NOT a property of the system.

Heat Transfer, Q¶

Q > 0: Heat transfer TO system
Q < 0: Heat transfer FROM system

Warning

Heat transfer is NOT a property of the system.

Rate of heat transfer¶

$Q = \int_{t_{1}}^{t_{2}}\dot{Q} \ dt\\ \text{Units: }W\text{ or }Js^{-1}$

Heat flux¶

Rate of heat transfer per unit area $\dot{q}$

$\dot{Q} = \int_{A}\dot{q} \ dA\\ \text{Units: }Wm^{-2}\text{ or }Js^{-1}m^{-2}$

Adiabatic process¶

Thermodynamic process involving no heat transfer with the surroundings

Conduction¶

Occurs within the same medium
Happens in solids, liquids, gases
Transfer of energy from energetic to less energetic particles

Fourier's Law¶

$\text{Heat transfer due to conduction, }\dot{Q}_{x} = -\kappa A\frac{dT}{dx}$

where:

$\kappa$ is the thermal conductivity of the material;
$A$ is the surface area of the material;
$\frac{dT}{dx}$ is the temperature gradient across the x-direction.

$\textbf{Assuming linear temperature gradient:}$

$\text{Heat transfer due to conduction, }\dot{Q}_{x} = -\kappa A\left(\frac{T_{2}-T_{1}}{L}\right)$

where:

$\kappa$ is the thermal conductivity of the material;
$A$ is the surface area of the material;
$T_{1}, T_{2}$ are the respective temperatures at two ends of the material;
$L$ is the length of the material.

Radiation¶

Happens in solids, liquids, gases
Emission due to changes in electronic configuration of material

Stefan-Boltzmann Law¶

$\text{Heat transfer due to radiation, }\dot{Q}_{e} = \epsilon\sigma AT_{b}^{4}$

where:

$\epsilon$ is the emissivity (radiation proportionality) of the material; $0\leq\epsilon\leq1$
$\sigma$ is the Stefan-Boltzmann constant
$A$ is the surface area of the material;
$T_{b}$ is the temperature of the emitting surface.

$\textbf{As such, net heat transfer due to radiation:}$

$\text{Net heat transfer due to radiation, }\dot{Q}_{e} = \epsilon\sigma A(T_{h}^4-T_{c}^4), \text{where }T_{h} > T_{c}$

where:

$\epsilon$ is the emissivity (radiation property) of the material; $0\leq\epsilon\leq1$
$\sigma$ is the Stefan-Boltzmann constant
$A$ is the surface area of the material;
$T_{h}, T_{c}$ are the temperature of the hot and cold surfaces respectively.

Convection¶

Occurs between solid and liquid; solid and gas

Newton's law of cooling¶

$\dot{Q}_{c} = hA(T_{h}-T_{c})$

where:

$h$ is the heat transfer coefficient;
$A$ is the surface area of the material;
$T_{h}, T_{c}$ are the temperature of the hot and cold surfaces respectively.

Heat transfer coefficient, h¶

Empirical parameter
Depends on flow pattern, fluid property, geometry
Forced convection
- Caused by external device (e.g. fan, pump)
- Larger h (more efficient)
Free/natural convection
- Caused by buoyancy effects (difference in air density)
- Smaller h (less efficient)

Laws of Thermodynamics¶

First Law of Thermodynamics¶

Energy is conserved.

$\Delta E = E_{2} - E_{1} = Q - W\\ \frac{dE}{dt} = \dot{Q}-\dot{W}\\ dE = \delta Q - \delta E$

Microscopic and Macroscopic Energy¶

$\Delta E = \Delta KE + \Delta PE + \delta U$

where:

$\Delta KE$ is the change in kinetic energy;
$\Delta PE$ is the change in potential energy;
the change in the above two energies happens at the macroscopic scale, i.e. changes in KE and PE can be seen;
$\Delta U$ is the change in internal energy;
the change in internal energy happens at the microscopic scale, i.e. changes in U cannot be seen.

Energy Balance¶

$\dot{E}_{in} + \dot{E}_{gen} -|\dot{E}_{out}| = \dot{E}_{st}$

where:

$\dot{E}_{in}$ is the rate of energy transfer in;
$\dot{E}_{gen}$ is the rate of energy generated;
$\dot{E}_{out}$ is the rate of energy transfer out;
$\dot{E}_{st}$ is the rate of energy stored.

Surface Energy Balance¶

No heat is generated or stored.

$\dot{E}_{in}-|\dot{E}_{out}| = 0$