# 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¶

1. Isolated system
• No exchange of energy
• No exchange of matter
2. Closed system
• Exchange of energy
• No exchange of matter
3. 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}

• 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.

• 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