Heat Transfer/Heat Balances

Macroscopic Heat Balances

The first law of thermodynamics states that energy cannot be created nor destroyed: doing an energy balance means imposing that the algebraic sum of the following variables is constantly zero in the considered system:

(1) the entering fluxes of heat;

(2) the exiting fluxes of heat;

(3) the electrical and/or mechanical energy generated or consumed by the system;

(4) sources of energy within the system (e.g. dynamite can be considered a source of energy if we analyze the dynamic of an explosion); (5) sinks of energy within the system (usually the ambient is considered as a sink of energy);

(6) the energy accumulated by the system during time.

When last term is zero, system is called "in steady-state conditions", or "steady" and its behavior does not depend from time; otherwise, it is called "unsteady" or "transient".

A macroscopic heat balance is a balance on an entire system, rather than an infinitesimal part of it. The ability to perform a balance on heat comes from the principle of conservation of energy, which tells us that energy is never created or destroyed. Since this is true, we can balance energy in the following manner:

Energy accumulated = Energy in - Energy out

To be completely general, we would have to include all forms of energy and energy changes in this balance: potential energy, kinetic energy, and internal energy, as well as all forms of heat and work. However, if there are heat effects present, and no shaft work is done, heat effects will usually be considerably larger than the effects of volumetric expansion, potential energy and kinetic energy changes, or electrical effects. In this case, the equation simplifies to:

Energy accumulated = Heat in - Heat out

Heat can come into a system either through the enthalpy of a mass stream or through conduction, convection, or radiation from an outside source. Heat leaves a system in the same manner. All accumulated energy will be in the form of internal energy, since potential and kinetic energy are neglected. Hence, the overall heat balance becomes:

$$ \frac{dU}{dt}= \sum_{k=1}^M \dot{m}_k\hat{H}_k + \sum_{j=1}^N \dot{Q}_j$$

This balance is useful for determining the total heat transfer necessary to accomplish a given temperature change in a system, since the internal energy and the enthalpy of any pure substance or mixture can be estimated as a function of temperature (given some experimental data) using thermodynamics. This approach does not require one to use the microscopic analysis if the inlet, outlet, and system temperatures are known (and preferably constant). The microscopic approach is useful for determining the temperature distribution and its change with time (if present) so that the temperature, and thus the energy requirements, can be evaluated at all time points.

If the system is at steady state, the accumulation term is zero so

$$ \sum_{k=1}^M \dot{m}_k\hat{H}_k + \sum_{j=1}^N \dot{Q}_j = 0$$

This implies that a closed system that does not work will only stay at steady state if no heat is transferred (neglecting potential and kinetic energy differences). This is similar in spirit Newton's first law: if an object is left alone, its velocity won't change.