# 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.

\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.

\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