Non-isothermal processes

The isothermal flow of incompressible liquid is described by equations (5.13) and (5.21), and the viscosity coefficient n = const. Hence, there are four equations for four unknowns - the pressure p and three velocity components u, v, and w. Thus, the system of equations is a closed one. For its solution it is necessary to formulate the initial and boundary conditions. Let us discuss now possible boundary conditions. Consider conditions at an interface between two mediums denoted as 1 and 2. The form and number of boundary conditions depends on whether the boundary surface is given or it should be found in the course of solution, and also from the accepted model of the continuum. Consider first the boundary between a non-viscous liquid and a solid body. Since the equations of motion of non-viscous liquid contain only first derivatives of the velocity, it is necessary to give one condition of the impermeability u i = u 2 at the boundary S, where u is the normal component of the velocity. The equations of motion of viscous liquid include the second-order derivatives, therefore at the boundary with a solid body it is necessary to assign two conditions following from the condition of sticking u i = u 2, Wii = u i where u is the tangential to S component of the velocity. If the boundary S is an interface between two different liquids or a liquid and a gas, then it is necessary to add the kinematic condition Ui = U2 = 

Processes accompanied by temperature change are called non-isothermal. The change of temperature may be caused by heat transfer through the medium s boundary 2, or heat production within the volume V of a liquid it may be the Joule heat from chemical reactions, viscous energy dissipation in the medium s motion, the change of medium s phase state (evaporation, condensation, fusion etc.). 

The thermal energy of a liquid particle is defined by the internal energy density, which depends on parameters of local thermodynamic condition. According to the first law of thermodynamics. 

Here we consider heat transport in the system due only to heat conductivity, without taking into account internal heat production due to Joule heat and/or the chemical reactions. 

Consider a liquid element of volume V enclosed by a surface S. Determine the full energy of the considered volume as sum of kinetic K and internal E energy, where 

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ASHRAE, Atlanta (1992)]. Process air stream 6, to be conditioned, passes through the adsorbent wheel, where it is dried. This is a non-isothermal process due to the release of heat of adsorption and transfer of heat from a wheel that may be above ambient temperature. The dry but heated air (7) is cooled in a heat exchanger that can be a thermal wheel. This stream (8) is further cooled, and the humidity adjusted back up to a comfort range by direct contact evaporative cooling to provide supply air. Regeneration air stream 1, which can be ambient air or exhausted air, is evaporatively cooled to provide a heat sink for the hot,... 

Figure 2.9. Approximate temperature ranges for densification processes contributing to structural evolution (non-isothermal processing with moderate ramp rate).

The design equation for this non-isothermal process may be written... 

Transfer of the theoretical results obtained for isothermal conditions to the non-isothermal case is rather simple provided the temperature dependencies of the constants are known. A theoretical analysis of non-isothermal processes may be limited by thermal runaway, which is equivalent to the thermal instability already thoroughly studied for chemical reactors.80,81 There are two limiting situations, known as the Semenov and Frank-Kamenetzky models 80,82 these correspond to periodic ideal stirred and periodic ideal plug reactors, respectively. 

Polymer synthesis is usually accompanied by a strong exothermal effect and, as a general rule, real process proceeds in non-isothermal conditions. This problem was considered in Section 2.6 for polyester synthesis, and the results discussed above are of generally applicable to all non-isothermal processes. 

Figure 2.32. Predicted dependences of a dimensionless induction period in a non-isothermal process on dimensionless shear rate at different values of the parameter i 0 (curve 1) 0.6 (curve 2) 1.2 (curve 3) 1.8 (curve 4).

The process temperature (TP) the initial temperature in the cooling failure scenario. In case of non-isothermal processes, the initial temperature will be taken at the instant when the cooling failure has the heaviest consequences (worst case). 

The media with which one has to deal when investigating preparation processes of hydrocarbon systems are invariably multi-phase and multi-component mixtures. Section II thus covers the aspects of the hydromechanics of physical and chemical processes necessary for an understanding of the more specialized material contained in following sections. Among these are transfer phenomena of momentum, heat, mass, and electrical charge conservation equations for isothermal and non-isothermal processes for multi-component and multi-phase mixtures equations of state, and basic phenomenological relationships. 

In previous works we have shown the great importance to control the working electrode temperature (non-isothermal process) and developed a new transfer function based on the sine wave modulation of the electrode temperature, (Olivier, Merienne, Chopart, and Aaboubi 1992). Thus the thermoelectrochemical (TEC) transfer function has been experimentally measured and compared with theoretical models for mass transport controlled systems or charge... 

In comparison to isothermal membrane processes, little attention has been paid to date to polarisation phenomena in non-isothermal processes. In non-isothermal processes such as membrane distillation and thermo-osmosis, transport through the membrane Occurs when a temperature difference is applied across the membrane. Temperature polarisation will occur in both membrane processes although both differ considerably in membrane structure, separation principle and practical-application. In a similar manner to concentration polarisation in pressure-driven membrane processes, coupled heat and mass transfer contribute towards temperature polarisation. 

Enthalpies of transition between different solid phases are in general determined by the eqnation similar to the equation (2.1). The inaccuracy of determination in the temperature and enthalpy of higher order solid-solid transitions will be larger than that of fusion due to the non-isothermal processes of higher order transitions. 

In most cases non-isothermal processes are used. A part of the energy of the reaction is used to heat the reactor to the desired process temperature in the so-called polytropic reaction mode. A typical temperature control strategy is to use an adiabatic phase, during which there is no cooling, so that the correct initiation of the reaction can be checked. Then, in a second stage, after reaching a specified temperature, the cooling system is switched. The correct choice of the initial temperature... 

When accounting for I, the constitutional equation holds as, A = (X,Xeq,T), and the true non-isothermal rate of chemical reaction, dWdt, becomes dependent on the sum da/dt Aty + adXg dt. Then the modified rate of a non-isothermal process, r, becomes more complicated [3,370,371,403] r = [dX/dt.. X dT/dt (dlnX, dTJ/X,/ = k(T) f(a). This expression, however, is of little practical use as the term, dXei/dt, tends to become zero for all near-equilibrium conditions. 

Segal Eugene (1933-) Rom. phys., expert in heterogeneous kinetics of non-isothermal processes (book Nonisothermal kinetics 1983)... 

As the number of components increases the solutions to the set of design equations, with their appropriate boundary conditions, become increasingly difficult and therefore it is usually necessary to make approximations. The mathematical models which are thereby obtained differ in their generality and in their applicability according to the extent and severity of the approximations which have been made. As an example of the number and type of assumptions and approximations which might need to be invoked for a non-isothermal process, let us consider the relatively simple model of Cooney (1974). In this case it is assumed that... 

The non-isothermal crystallisation of PETP was examined by temp, modulated DSC(TMDSC). A new analytical model of TMDSC was applied to the process, taking account of the response of exothermic heat flow to temp, modulation in an apparent heat capacity of complex quantity. By examining the frequency dependence of the apparent heat capacity, the applicability was successfully examined for the non-isothermal process. The method was capable of determining the temp, dependence of crystal growth rate from TMDSC data analysis. The results were in good agreement with the dependence determined from literature values of spherulite growth rate measured by optical microscopy. 12 refs. 

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