*2.1. Theoretical Development*

For a reactor without any HTC reaction (i.e., without any feedstock inside reactor), we can estimate the HTC autogenic pressure with that of pure water at the HTC reaction temperature. This information is often visualized for hydrothermal systems with a pressure-temperature (*P-T*) phase diagram for water, showing the regions for the different types of processes, e.g., gasification, liquefaction, carbonization. However, to help us understand the process conditions during a hydrothermal reaction, the use of the temperature-volume (*T*-*v*) phase diagram for water is a powerful tool which provides information on *P* and *T* as well as the distribution of water between phases as a function of the overall specific volume of water (liquid and steam) in the reactor *vR* (Figure 1). Using this diagram, one can understand the thermodynamic equilibrium at the chosen process conditions of the reactor system, e.g., temperature, pressure and mass of water in the system. In Figure 1, the saturation line represents the boundary condition for the phase change. For most HTC/VTC reactor systems, the reaction zone is usually located within the saturation curve, where steam and liquid phases coexist. The operating path for a batch system can be followed from the starting process conditions until the target conditions are met and the holding time begins. The closer the target point is to the steam or liquid saturation lines, the higher the amount of that phase. Since a log scale is used for the x-axis, the ratio between the two phases cannot easily be determined visually from the figure. The calculation procedure is developed in the following section.

**Figure 1.** Temperature-volume (*T*-*v*) diagram for water showing the common operating region for vapor and hydrothermal carbonization reactions.

Total mass of water in the reactor is

$$M\_{H2O} = \text{x}\_{L}M\_{H2O} + \text{x}\_{V}M\_{H2O} \tag{1}$$

where

*MH2O* = total mass of liquid and vapor water in the reactor (kg);

*xL* = mass fraction of liquid water;

*xV* = mass fraction of vapor water (or steam quality);

$$
\mathbf{x}\_L + \mathbf{x}\_V = 1.
$$

As the HTC reactor is heated beyond the boiling temperature, the reactor volume is mostly filled with liquid water and steam, and the following relationship can be developed assuming both liquid water and steam are in equilibrium (i.e., for the saturated liquid–vapor region; Figure 1).

$$V\_R = \mathbf{x}\_L M\_{H2O} \upsilon\_L + \mathbf{x}\_V M\_{H2O} \upsilon\_V \tag{2}$$

where

*VR* = reactor volume (m3);

*vL* = specific volume of saturated liquid water (m3/kg);

*vV* = specific volume of saturated steam (m3/kg).

Combining Equations (1) and (2), the mass fraction of vapor water (*xv*) can be calculated by knowing the thermophysical properties of water at those conditions, the mass of water in the reactor and the reactor volume:

$$\mathbf{x}\_V = \frac{\upsilon\_R - \upsilon\_L}{\upsilon\_V - \upsilon\_L} \tag{3}$$

where

*vR* = *VR*/*MH2O,* overall specific volume of reactor water and steam mixture (m3/kg).

As the liquid-steam mixture in the reactor is heated, the volume of liquid water expands due to the decrease in water density ρ*L*. Using Equation (3) along with values for saturated vapor and liquid specific volumes [16], the fraction of liquid-water occupying the reactor volume *VFw* can be estimated at the HTC reaction temperature:

$$VF\_w = \frac{V\_w}{V\_R} = \frac{(1 - \chi\_V)\upsilon\_L}{\upsilon\_R} \tag{4}$$

where

*Vw* = volume of liquid water in the reactor at temperature *T* (m3);

*VFw* = volume fraction of liquid water in the reactor at temperature *T* (-).

As long as the reactor volume is larger than the bulk liquid water volume (i.e., *VFw* < 1), we can assume the liquid and vapor water phases are in equilibrium and the autogenic pressure can be estimated from the saturation properties of water using saturated steam tables [16–18].

If the temperature is further increased so that the liquid volume completely fills the reactor due to the decrease in its density (i.e., *VFw* = 1, and *xL* = 1), the liquid water will enter the subcooled liquid compression region. This region can be seen in the *T*-*v* phase diagram, left of the saturated vapor curve (Figure 1). There is no longer any headspace in the reactor and the water density in the reactor system at this point (also called overall reactor water density) becomes constant and can be calculated from *D* = *MH*<sup>2</sup>*O*/*VR*. As the rigid reactor walls are suppressing the tendency of the liquid volume to increase in response to the decrease in liquid water density, the reactor pressure increases rapidly as the water expands with the increase in temperature. When *VFw* > 1 calculated from Equation (4) (i.e., physically impossible unless the reactor explodes), the reactor pressure in this range can be estimated with liquid compressibility factor for subcooled water:

$$P = Z\_L \times D \times RT / MW\_{H2O} \tag{5}$$

where

*P* = reactor pressure (MPa);

*ZL* = liquid compressibility factor for subcooled water (-);

*D* = overall reactor water density, *MH*<sup>2</sup>*O*/*VR* (kg/m3);

*<sup>R</sup>* <sup>=</sup> universal gas constant (8.31451 <sup>×</sup> <sup>10</sup>−<sup>3</sup> <sup>m</sup>3-MPa/kmol-K);

*T* = reactor temperature (K);

*MWH2O* = molecular weight of water (kg/kmol).

To illustrate the danger of a potential reactor explosion if the liquid fills the reactor completely, example calculations to estimate the reactor pressure at three common HTC temperatures using Equation (5) are reported in Table 1. The values of liquid compressibility factor of the subcooled water reported by Lemmon et al. (2018) were used. A value for *D* was chosen that is slightly higher than the saturated liquid water density at 200 ◦C. This simulates the reactor pressure for the case when the liquid water fills the reactor completely at 200 ◦C. A further increase in *T* to 250 ◦C will rapidly increase P from 2 to 81.6 MPa, a pressure that many HTC reactors are not made to withstand. For instance, maximum allowable pressures for common laboratory reactors range from 13.3 to 34.5 MPa [19]. In contrast, if there is less liquid water added and more headspace in the reactor so that the liquid water-vapor equilibrium can exist at all operating temperatures, the pressure increase would follow the saturation pressure, increasing only from 1.6 to 4.0 MPa.

In order to avoid the subcooled compressible region, some manufacturers of pressure equipment recommend calculating the maximum allowable water mass using a safety factor and the ratio of ρ*<sup>L</sup>* or its inverse *v*<sup>L</sup> at the desired *T* to that at room temperature [20]. It is also important to note that the actual HTC pressure will be higher than that from the pure water because of gas production (predominantly CO2) from HTC reactions.


**Table 1.** Example calculations for the effect of increasing the hydrothermal carbonization (HTC) temperature on reactor pressure for *D* = 865 kg/m<sup>3</sup> (values for saturated water properties taken from Lemmon et al. (2018).
