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# 1.5 Energy balance

The energy flow goes from the warm medium to the cold medium through the heat transfer area of the BPHE. In addition to the size of the heat transfer area, the amount of energy transported also depends on the heat transfer coefficient and the temperature difference between the two sides. This relation is described in the heat transfer equation (eq. 2): ## Area (A)

Increasing the area of a heat exchanger implies that more energy can be transferred. For BPHEs, a larger area can be achieved by increasing the size and/or number of plates, which means more stainless steel and copper brazing. Hence, increasing the surface area implies higher costs.

## Heat transfer coefficient (k)

Energy may be transported from a hot fluid to a colder fluid in three ways:

1. Conduction – The heat is conducted through solid material or a stationary liquid. In the stainless steel walls of a heat exchanger and in laminar flow (slow moving) regions, heat is transported only by conduction. The conductivity varies with the physical properties of the medium.
2. Convection – Movements of the fluid itself also transport energy. Turbulently flowing media and boiling/condensing fluids are very agitated, and will therefore transport energy mostly by convection.
3. Radiation – For very hot surfaces (T > 1000°C), electromagnetic radiation will become the most important means of heat transport. Radiation does not contribute significantly to the heat t​ransfer in BPHEs, due to their considerably lower working temperature.

In BPHEs, the energy is therefore transferred through conduction and convection, and examples of these types of energy transport are shown in In BPHEs, the energy is therefore transferred through conduction and convection, and examples of these types of energy transport are shown in Figure 1.4.

The space between the dotted lines and the wall in Figure 1.5 is often called the film thickness. The heat transfer rate within the film is significantly lower than in the bulk liquid, because the temperature gradient decreases dramatically in this area (see Figure 1.5). The reason for the poorer heat transfer is the laminar flow that is always obtained near a plane wall. Laminar flow does not transfer energy as well as turbulent flow. The overall heat transfer coefficient (k) describes the total effect of conduction and convection on the energy transfer: The essence of equation 3 is that a high film coefficient and thermal conductivity and a thin plate lead to a high k-value. Thermal conductivity is a material-specific constant, and the film coefficient is a measure of how well heat is transferred by a specific fluid. For turbulent flows, α is always higher than for laminar flows.

With a higher overall heat transfer coefficient (k), more energy can be transferred per heat transfer area. Because this leads to a more costeffective heat exchanger, it is very important to improve the k-value by all means possible.

## Temperature difference (dT)

The temperature difference between the hot and cold media is the driving force in energy transfer. A large temperature difference means that a smaller heat transfer area and/or a smaller heat transfer coefficient may be used to achieve the same energy transfer. It is therefore important to try to maximize the temperature difference between the hot and cold sides.

Figure 1.6 shows a single-phase temperature profile through a BPHE.

Because the temperature difference between the hot and cold sides varies through the heat exchanger, the logarithmic mean temperature difference (LMTD) is used. The definition of LMTD is shown in equation 4: Please note that the logarithmic mean temperature difference (LMTD) may be used only for single-phase calculations (see chapter 6.10).

## Preservation of energy

The energy in a liquid flow can be described with the following formula: Note that equation 5 is valid only for one-phase heat exchange. The specific heat capacity can be interpreted as the amount of energy required to increase the temperature of 1 kg liquid by 1°C at constant pressure. The specific heat capacity varies for different liquids and different temperatures.

Equations 2 and 5 together describe the preservation of energy inside a BPHE, which is shown in equation 6. This equation, as well as Figure 1.7, indicates that there are no theoretical heat losses to the surroundings in a BPHE. 