# Heat [Q]

Heat is not a property. Energy transferred across the boundary of a system in the form of heat always results from a difference in temperature between the system and its immediate surroundings. Energy transfer resulted from a temperature difference is what separates heat transfer from work. We will not consider the mode of heat transfer, whether by conduction, convection or radiation, thus the quantity of heat transferred during any process will either be specified or evaluated as the unknown of the energy equation.

By convention:

$$Q>0: \quad \text{heat transfer into the system}$$

$$Q<0: \quad \text{heat transfer out of the system}$$ This sign convention will be used throughout the course. Notice this convention is the opposite of that given for work!!

A differential quantity of heat $\delta Q$ may be integrated over a process to find the total amount of heat transfer during that process from state (1) to a final state (2), but note that the limits on the integral do not represent the value of heat at the states, for heat cannot be measured. Do not be tempted to evaluate $Q_{2}-Q_{1}$

$$Q=\int_{1}^{2} \delta Q$$

Only the rate heat transfer $\dot{Q}$ can be measured. The amount of energy transfered by heat can be found by integrating $\dot{Q}$ over time:

$$Q=\int_{t_1}^{t_2} \dot{Q} dt.$$

In the absence of heat transfer, a process is said to be adiabatic.

## Heat transfer modes

Conduction

Conduction heat transfer occurs in substance which are relatively still, but can occur in solids, liquids, and gases. Conduction relies on energy transfer between neighbor particles. The efficacy with which particles transport heat, or molecular motion, from one to another is quantified by a material’s thermal conductivity $\kappa$. We use Fourier’s law to determine the rate of heat transfer via conduction through a static material:

$$\dot{Q} = -\kappa A \frac{dT}{dx}$$

Thus, greater thermal conductivity, contact areas, and temperature gradients all contribute to increased conduction heat transfer.

Convection

Convection is the transfer of heat between a solid surface and adjacent moving fluid. We quantify convection with Newton’s law of cooling:

$$\dot{Q} = hA(T_\text{s}-T_\text{f}),$$

where h is the heat transfer coefficient that depends on the type of fluid and flow conditions, A is area, and $T_\text{s}$ and $T_\text{f}$ and the surface and fluid bulk temperatures respectively.