Analytical Modeling and Computer Simulation of Heat Transfer Phenomena during Hydrothermal Processing Using SOLIDWORKS ®

Mathematical modeling of unit operations especially in biomaterials processing plays an important part in understanding simultaneous heat and mass transfer and fluid flow phenomena. Hydrothermal/supercritical processes are one such process which utilizes high temperature and pressure to synthesize materials for varied applications. The present study constitutes development of a model incorporating all standard transport equations and transient conditions to predict the behavior of process and thus materials upon heating to high temperature in an enclosed composite (Steel and Teflon (PFA) vessel. Commercial software package SOLIDWORKS® is employed to simulate and output is presented as animations after post-processing. Results yield very useful data and information about process; materials of construction and materials being processed which helps in optimization.

Engineering critical temperature of the water (T c = 647 K & P c = 221 bar). Precise calculation of the heat generation and transfer can give more control over the hydrothermal process. Any hydrothermal process involves two steps i.e. heat generation and heat transfer. Heat generation may be attained via electrical power [9] [10]. Heat transfer depends upon the process being carried out, type of insulation and configuration of the hydrothermal reactor. For a successful modelling, it is important to incorporate all temperature & pressure dependent material properties.
External heating elements are placed for more precise control of temperature during the hydrothermal treatment. The heat generation (Q) in joules is in accordance with the Joule heating law (Q = I 2 Rt), where I is the current in amperes, R is the resistance of the heating elements and t is the duration in seconds.
Heating elements may be constructed through high electrical resistance materials (stainless steels, tungsten alloys & nickel chromium alloys).
Joule heating is also known as Ohmic heating and it can be thought of the similar heating process used by microwaves but at different frequencies. The main advantage of the Joule heating is the uniformity and better control in a hydrothermal process. It depends upon the electrical resistance of the heating elements, electrical field strength, residence time and type of waveforms. For the safe and reliable operation, it is important to understand factors on which heat generation depends and modelling is efficient way to understand the process. Some advantages of the Joule heating are homogeneity, better control, less time, more efficient and environmentally friendly [9] [10]. Second step involves heat transfer process that depends upon the process being carried out inside the reactor and material properties [11] [12]. To avoid the loss of the heat and for better control over the process, usually such hydrothermal reactors are lined with some insulating layer such as Teflon. The heat transfer process is analyzed via standard transport equations [13] [14]. The main challenges involved during modelling of the hydrothermal operations are lack of material thermal properties at various temperatures and pressures, adiabatic increase or decrease in the temperature due to local pressure variation. Incorporation of heat transfer coefficient can result in inaccuracies in the model and measures have been adopted to counter this. The main purpose of the study is to develop the numerical simulation for the heat generation and transfer process during hydrothermal operation. It aims to develop a computational platform for the heat control (mainly for generation and transfer). The process is analyzed via mathematical modelling and verified through simulation using commercial code (SOLIDWORKS®). The heat generation is calculated via Joule heating law and heat transfer process is modelled via standard transport equations [13] [14]. Previous studies have incorporated only single value of the heat transfer coefficient for the heat transfer throughout the process, however in real operation, it is a transient in nature [14]. The major contribution of this work is the incorporation of the transient heat transfer coefficients, which are calculated in each iteration. It provides more accurate model and precise values of output temperature. A model is developed and implemented via iterative approach via constitutive equations for the calcu-lations of heat transfer coefficients during each step, based upon the correlations developed for conduction, convection, and radiation [14]. Heat generation and heat transfer simulations are carried out in SOLIDWORKS® software package for verification purposes. A computer simulation-based approach is developed for the accurate prediction of heat transfer during the hydrothermal process.

Mathematical Formulation
To formulate a generalized mathematical model corresponding to heat transfer phenomena during hydrothermal processing in sealed vessels, the following assumptions were made: 1) Main source of heating is electrical resistance heating in a Box type furnace.
Electric heater is at constant temperature.
2) Thermal properties of mold are not constant & do change with change of temperature.
3) Thermal conductivity of mold (steel) is very high and that of TEFLON is very low. 4) Rate of heat transfer by convection and radiation is negligible.
5) The furnace is very well insulated. 6) And most importantly, this model will consider the incorporation of the transient heat transfer coefficient during the modelling process.
Based upon these assumptions, following is considered The steel and TEFLON (composite) vessel is considered identical to sealed pressurized reactor/vessel in which quantity of heat to and from system is zero (Adiabatic process).

Heat transfer for Hydrothermal Reactor
In order to formulate a mathematical model for heat transfer problem during heating, consider a stainless-steel engulfed TEFLON vessel placed inside an electric furnace (resistance heating).
There are three steps in which heat is transferred: 1) First step in which electrical power is used to heat the resistance element of furnace to desired temperature.
2) Second step in which heat energy generated in step one is used to heat the vessel (steel).

3) Third step in which heated steel vessel heats the TEFLON vessel.
Time of heating is determined separately during each interval using different relations and then individual times are summed up together to get total time taken for heating. All the heat generated in this step goes into overcoming material heat losses expressed in term of specific heat capacity of substance when it is heated across a temperature difference [14]. This may be written as.

First Step of Heating-Heat Generation
where m = mass of heating element (kg). C p = specific heat capacity of material (function of temperature) (KJ/Kg•K). T 2 = final temperature (K). T 1 = initial temperate (K).
This is the time in which elements reach the set operating temperature (T 2 ) from initial temperature (T 1 ). Putting back the calculated time in (2) yields energy (heat) generated (hence available for transfer to raise the temperature of vessel).
Various factors affect time of heating during first step such as thermal conductivity of mold material [12] [15] [16], heating conditions, maximum power rating of furnace [9], safe operating limit, specific heat of metal, etc. All these should be taken into account while designing a heating system [10]. Soon after the element is heated to a designated temperature, second step of heating (heat transfer) begins.

Second Step of Heating-Heat Transfer
When resistance heating element is heated to its set operating temperature, the heat it produces (using electrical power) goes into raising the temperature of hydrothermal vessel [ [14].
Energy (heat) generated once again may be written as This is the same as (2) above, with calculated time value from (5). This heat is responsible for raising the outside and then inside temperature of vessel. Its quantity keeps on changing (raising) with time as more and more time (which will become visible later) is needed to bring the inside temperature to reaction temperature. However, its average value remains constant (actually used in calculations), by automatic control of furnace. When vessel is placed in furnace, the outer (infinitesimal) skull of stainless steel body is instantaneously heated to high temperature (Reaction temperature, temperature of furnace chamber). Heat transfer also starts subsequently and temperature of reactor body as a whole starts increasing. As a result of this, temperature of furnace chamber starts decreasing. This is compensated by automatic switching ON and OFF of furnace elements. This in turn depends on the programming of furnace (not discussed here) such that every time chamber temperature raises a certain degrees above set operating temperature, Furnace shuts OFF, remain shut OFF, until the chamber temperature drops below another designated temperature, when it automatically powers ON and starts heating. In this way, the average temperature (thus heat produced) of furnace remains constant. The reliability of furnace to function satisfactorily between these limits depends on factors such as materials of construction of elements (type (KANTHAL, Nichrome, SiC and Stainless Steel), quality, geometry and amount), insulation (type, quality, geometry & amount), type & materials of construction of furnace, temperature reading, sensing & reporting devices (thermocouples, sensors, indicators, displays) i-e furnace will transfer the heat positively for longer period of time even in shut OFF condition (thus saves electricity) if aforementioned variables are of optimized value.
Heat transfer from high temperature T 2 to lower temperature T 1 may be represented by following equation where T 2 and T 1 are outside and inside temperatures (K) of stainless steel vessel at infinitesimal value of time t and R t is the thermal resistance of materials (K/W) at the same time.

Determination of Thermal Resistance
This thermal resistance can further be expressed in the form [14] t R L kA = (8) where L = thickness of mold material(s) (m). A = Surface area exposed/available for heat transfer (m 2 ). k = Thermal conductivity of material (W/m-K) (strong function of temperature). For stainless steel (304) k may be expressed by general relation 6 2 9.705 0.0176 1.60 10 where T = temperature of operation (K) at infinitesimal time t.
This thermal conductivity is different for different materials and its temperature dependence is also different for different materials. A huge collection of literature is available to find it for different materials at different temperatures.

Results and Discussion
Results

Effect of Metal Temperature
When simulation of temperature of vessel and its variation across vessel wall thickness performed as a function of time, following pattern is obtained which is shown in Figure 1. As can be seen when outside wall of vessel is heated un-  middle. Furthermore, the material inside the vessel will also not get heated to its desired value as most of its mass is below horizontal half plane and in this configuration, maximum heat will never reach there. The variation in the temperature with respect to various conditions is shown in Figures 1(a)

Defects
Six types of defects are identified during simulation which may give rise to poor results as shown in Figure 5.  2) Improper surface selection.
3) Convection from outside of base (improper base insulation/protection) (cold spot outside as well as inside).

Conclusions
Inverse relationship is found between time and inside/outside temperature as shown in the following Table 1.