Nonisothermal Flows. Temperature Equation

Rheological equation in the nonisothermal case. The most general onedimensional nonisothermal equation of state for non-Newtonian fluids can be written in the form F(t, 7, T) = 0, where r is the tangential stress, 7 is the shear rate, and T is the dimensional temperature. Some special types of the equation of state are presented in Tables 6.1 and 6.3, where the rheological parameters n, A, B, C, Ho, Hog, and To must be treated as functions of temperature T. 

Let us consider power-law fluids in more detail. Experiments [141] show that the index n of non-Newtonian behavior of a substance may be treated as a constant if the temperature differences in the flow region do not exceed 30 to 50 K. The medium consistence k = k(T) is much more sensitive to temperature inhomogeneities and decreases with increasing T. Therefore, the rheological equation of state for a power-law fluid in the nonisothermal case can be written as follows  

Now let us show that in the nonisothermal motion of fluid in tubes and channels some critical phenomena may occur related to the existence of a maximum admissible pressure gradient. Once this value is exceeded, the steady-state flow pattern is violated. This is accompanied by an accelerated decrease in the apparent viscosity and increase in the fluid velocity. This phenomenon is known as the hydrodynamic thermal explosion [52] and is caused by the nonlinear dependence of the apparent viscosity on temperature. Specifically, under certain 

In what follows we assume that the thermal conductivity coefficient of the medium is independent of temperature. 

Equation for the temperature distribution. In the region of hydrodynamic and thermal stabilization, nonisothermal rectilinear steady-state flow of a power-law fluid in a circular tube of radius a at constant temperature on the tube surface is described by Eqs. (6.4.1), (6.5.7), and (6.6.1). We assume that the no-slip condition is satisfied on the tube wall, and the boundary conditions for temperature are given in (6.5.8). 

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In nonisothermal flow, temperature variation induces density variation. Boussinesq assumed that density varies little with temperature and can be assumed as an average density with the exception of buoyancy term. However, the scaling analysis has shown that the buoyancy term can be neglected for a microsystem. Therefore, the simplified N-S equation is written as... 

Further the pressure and temperature dependences of all the transport coefficients involved have to be specified. The solution of the equations of change consistent with this additional information then gives the pressure, velocity, and temperature distributions in the system. A number of solutions of idealized problems of interest to chemical engineers may be found in the work of Schlichting (SI) there viscous-flow problems, nonisothermal-flow problems, and boundary-layer problems are discussed. 

The equations which govern the nonisothermal flow of a reactive fluid are derived in several texts on transport phenomena and polymer processing (e.g. refs. 1,2). Regarding velocity, temperature, and concentration of unreacted species as the fundamental variables, the governing equations can be written as ... 

Here, u represents a characteristic velocity of the flow and usotmd is the speed of sound in the fluid at the same temperature and pressure. It may be noted that usound for air at room temperature and atmospheric pressure is approximately 300 m/s, whereas the same quantity for liquids such as water at 20°C is approximately 1500 m/s. Thus the motion of liquids will, in practice, rarely ever be influenced by compressibility effects. For nonisothermal systems, the density will vary with the temperature, and this can be quite important because it is the source of buoyancy-driven motions, which are known as natural convection flows. Even in this case, however, it is frequently possible to neglect the variations of density in the continuity equation. We will return to this issue of how to treat the density in nonisothermal flows later in the book. 

The great virtue of this model is that unlike other models, it explicitly incorporates the dependence of viscosity on temperature as well as shear rate through the temperature dependence of the zero-shear viscosity (to be discussed shortly). Equation 14.8 is written to emphasize that point. This makes it particularly well suited for nonisothermal flow calculations. 

The centerline temperature differential within the zone of fully established turbulent flow (Zone 3) of a nonisothermal jet can be derived using equations of momentum (Eq. (7.39)) and excessive heat conservation along the jet <. S... 

Isothermal jets are not influenced much by small changes in flow rate the size of the influence can be seen from the different equations. Nonisothermal jets could be changed substantially by small differences in both outlet velocity (flow rate) and/or temperature. 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

Nonisothermal reaction-diffusion systems represent open, nonequilibrium systems with thermodynamic forces of temperature gradient, chemical potential gradient, and affinity. The dissipation function or the rate of entropy production can be used to identify the conjugate forces and flows to establish linear phenomenological equations. For a multicomponent fluid system under mechanical equilibrium with n species and A r number of chemical reactions, the dissipation function 1 is... 

The parameters of the model are the activation energies for viscous flow (AE ) and reaction rate ( Ej ) their respective pre-exponential erms. This equation provides a predictive, analytical expression with which one can model nonisothermal cure using any appropriate time-temperature function [T(t)] one chooses, appropriate to the curing process of interest. For example, the baking of a coated substrate in an oven may be modeled by a relaxation-type heating function, with time constant (t) to take account of the thermal inertia of the substrate. (Eq. 4)... 

Here, the last two equations define the flow rate and the mean residence time, respectively. This formulation is an optimal control problem, where the control profiles are q a), f(a), and r(a). The solution to this problem will give us a lower bound on the objective function for the nonisothermal reactor network along with the optimal temperature and mixing profiles. Similar to the isothermal formulation (P3), we discretize (P6) based on orthogonal collocation (Cuthrell and Biegler, 1987) on finite elements, as the differential equations can no longer be solved offline. This type of discretization leads to a reactor network more... 

Nonuniform temperatures, or a temperature level different from that of the surroundings, are common in operating reactors. The temperature may be varied deliberately to achieve optimum rates of reaction, or high heats of reaction and limited heat-transfer rates may cause unintended nonisothermal conditions. Reactor design is usually sensitive to small temperature changes because of the exponential effect of temperature on the rate (the Arrhenius equation). The temperature profile, or history, in a reactor is established by an energy balance such as those presented in Chap. 3 for ideal batch and flow reactors. 

There are a multitude of variations for semibatch operation. Equation (5-22) already includes restrictions that limit its application to specific operating conditions for example, constant mass-flow rates. A frequently encountered case for nonisothermal operation is one in which there is no product stream, one reactant is present in the reactor, and the temperature is controlled by the flow rate of the feed stream containing the second reactant. Figure 4-17 shows this type, and Example 4-13 illustrates the design calculations for isothermal operatioh. The energy balance for this situation reduces to... 

The design formulation of nonisothermal plug-flow reactors consists of /+2 nonlinear first-order differential equations. Note that usually the inlet temperature of the heating/cooling fluid, Tp, is known. Hence, the case of co-current... 

For nonisothermal recycle reactors, we have to incorporate the energy balance equation to express variation in the reactor temperature. The energy balance equation is the same as that of a plug-flow reactor. 

Consider a two-phase nonisothermal turbulent flow in which droplets move under the influence of fluid drag force and their temperature, Tj, changes due to evaporation and the thermal interaction (driven by the temperature difference, T — Td) with the carrier fluid. Here, T is the temperature of the fluid in the vicinity of the droplet. The rate of evaporation governs the size (diameter) of the droplets. A variety of equilibrium and nonequilibrium evaporation models available in the literature were recently evaluated by. Miller et al. [16]. Here, the model which was used in the previous DNS work is selected [17]. The Lagrangian equations governing the time variation of the position X. velocity V, temperature Td, and diameter dd of the droplet at time t can be written as... 

Adiabatic or nonisothermal operation of a stirred tank reactor presents a different physical situation from that for plug flow, since spatial variations of concentration and temperature do not exist. Rather, reaction heat effects manifest themselves by establishing a temperature level within the CSTR that differs from that of the feed. Thus, when we use the terms adiabatic or nonisothermal in reference to CSTR systems, it will be understood to imply analysis where thermal effects are included in the conservation equations but not to imply the existence of thermal gradients. 

For noncatalytic homogeneous reactions, a tubular reactor is widely used because it cai handle liquid or vapor feeds, with or without phase change in the reactor. The PFR model i usually adequate for the tubular reactor if the flow is turbulent and if it can be assumed tha when a phase change occurs in the reactor, the reaction takes place predominantly in one o the two phases. The simplest thermal modes are isothermal and adiabatic. The nonadiabatic nonisothermal mode is generally handled by a specified temperature profile or by heat transfer to or from some specified heat source or sink and a corresponding heat-transfer area and overall heat transfer coefficient. Either a fractional conversion of a limiting reactant or a reactoi volume is specified. The calculations require the solution of ordinary differential equations. 

The temperature varies with the reaction time in the nonisothermal batch reactor. To perform the energy balance, we use the same energy balance equation 14.67, annulling the molar flow terms, but considering the variation of sensible heat with temperature and time. Then,... 

As an example of the application of equilibrium theory to nonisothermal systems we consider here a plug flow system, with one adsorbable component, in which the concentration of adsorbable species and the temperature changes are both small enough to validate the constant velocity approximation. For such a system the differential mass and heat balance equations are... 

The interpretation of Equations 3.95 and 3.96, as well as Equations 3.97 and 3.98, is simple if the number of molecules in the chemical reaction increases, the expression attains a positive value (since > 0 and > 0). Simultaneously, the terms in parentheses, in Equations 3.95 through 3.98, become larger than unity (>1) inside the reactor. Under isothermal conditions (T = To), the volumetric flow rate inside the reactor is thus increased. Under nonisothermal conditions—with strongly exothermic or endothermic reactions— the temperature effect, that is, the term T/Tq, results in a considerable change in the volumetric flow rate. 

In developing Eqns. (3-26)-(3-28), we assumed that the temperature was constant in any cross section normal to the direction of flow. We did not assume that the temperature was constant in the direction of flow. For a PFR, the reactor is said to be isothermal if the temperature does not vary with position in the direction of flow, e.g., with axial position in a tubular reactor. On the other hand, for nonisothermal operation, the temperature will vary with axial position. Consequently, the rate constant and perhaps other parameters in the rate equation such as an equilibrium constant will also vary with axial position. The design equations for an ideal PFR are valid for both isothermal and nonisothermal operation. 

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