Isothermal and adiabatic flow

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

Lawlb. C. . Trans. Am. Inst. Chem. Eng. 39 (1948) 385. Isothermal and adiabatic flow of compressible fluids. 

This is the basic differential equation that is to be integrated. To do this the relation between V and p must be known so that the integral of dp/V can be evaluated. This integral depends upon the nature of the flow and two important conditions used are isothermal and adiabatic flow in pipes. 

Compressible fluid flow occurs between the two extremes of isothermal and adiabatic conditions. For adiabatic flow the temperature decreases (normally) for decreases in pressure, and the condition is represented by p V (k) = constant. Adiabatic flow is often assumed in short and well-insulated pipe, supporting the assumption that no heat is transferred to or from the pipe contents, except for the small heat generated by fricdon during flow. Isothermal pVa = constant temperature, and is the mechanism usually (not always) assumed for most process piping design. This is in reality close to actual conditions for many process and utility service applications. 

Compressibility of a gas flowing in a pipe can have significant effect on the relation between flowrate and the pressures at the two ends. Changes in fluid density can arise as a result of changes in either temperature or pressure, or in both, and the flow will be affected by the rate of heat transfer between the pipe and the surroundings. Two limiting cases of particular interest are for isothermal and adiabatic conditions. 

Because the gas viscosity is not highly sensitive to pressure, for isothermal flow the Reynolds number and hence the friction factor will be very nearly constant along the pipe. For adiabatic flow, the viscosity may change as the temperature changes, but these changes are usually small. Equation (9-15) is valid for any prescribed conditions, and we will apply it to an ideal gas in both isothermal and adiabatic (isentropic) flow. 

In the case of adiabatic flow we use Eqs. (9-1) and (9-3) to eliminate density and temperature from Eq. (9-15). This can be called the locally isentropic approach, because the friction loss is still included in the energy balance. Actual flow conditions are often somewhere between isothermal and adiabatic, in which case the flow behavior can be described by the isentropic equations, with the isentropic constant k replaced by a polytropic constant (or isentropic exponent ) y, where 1 < y < k, as is done for compressors. (The isothermal condition corresponds to y= 1, whereas truly isentropic flow corresponds to y = k.) This same approach can be used for some non-ideal gases by using a variable isentropic exponent for k (e.g., for steam, see Fig. C-l). 

The differential energy balances of Eqs. (6.10) and (6.15) with the friction term of Eq. (6.18) can be integrated for compressible fluid flow under certain restrictions. Three cases of particular importance are of isentropic or isothermal or adiabatic flows. Equations will be developed for them for ideal gases, and the procedure for nonidcal gases also will be indicated. 

Knowledge of these types of reactors is important because some industrial reactors approach the idealized types or may be simulated by a number of ideal reactors. In this chapter, we will review the above reactors and their applications in the chemical process industries. Additionally, multiphase reactors such as the fixed and fluidized beds are reviewed. In Chapter 5, the numerical method of analysis will be used to model the concentration-time profiles of various reactions in a batch reactor, and provide sizing of the batch, semi-batch, continuous flow stirred tank, and plug flow reactors for both isothermal and adiabatic conditions. 

The relationship between the inlet and outlet temperature depends on the heat exchange between the fluid flowing through the bed and the surrounding fluid and/or the reactor parts. The rate of this heat exchange depends on the reactor design and is difficult to predict theoretically. The two limiting situations of isothermal and adiabatic operations can be considered in the evaluation of the reactor performance. Under isothermal conditions,... 

Figure 11.22 Comparison of plug-flow isothermal and adiabatic palladium membrane reactors for dehydrogenation of 1-butene [Itoh and Govind, 1989]...

Membrane-enclosed packed-bed reactor. Plug-flow regime at both membrane sides. Both isothermal and adiabatic conditions considered. No axial or radial diffusion. Negligible pressure drop at the catalyst side. 

For comparison, for adiabatic plug-flow reactor (R = 0) of the same volume, the outlet extent is Zout — 0.742, and 9out =1.135. The production rate of product B is 1,131 mol/min. For adiabatic CSTR R = oo) of the same volume, the outlet extent is Zout = 0.841, and 9out= 1.132. The production rate of product B is 1260 mol/min. Note that for both isothermal and adiabatic operation, a recycle reactor provides a higher production rate of product B than a corresponding plug-flow reactor and a CSTR. 

It was pointed out by Young and Finlayson [55] that for reactors with finite wall heat transfer, the ultimate conditions for a large reactor will be determined by the heat exchange, whereas in isothermal and adiabatic situations the concentrations and temperatures are determined only by the reaction, and so for the latter case, the differences between plug flow and axial dispersion model results always diminish with increasing reactor length. But, differences between the two models at the entrance may persist for reactors with wall heat transfer. They provide alternate criteria for the unimportance of axial dispersion (of both mass and heat) effects. Mears [59] pointed out that these new criteria were not general, and provided alternate ones for equal feed and wall temperatures. The new criteria are... 

J. M. Castro, S. D. Lipshitz, and C. W. Macosko [AIChE J., 28, 973 (1982)] modeled a thermosetting polymerization reaction in a laminar flow reactor under several different operating conditions. Demonstrate your ability to simulate the performance of a plug flow reactor for this reaction under both isothermal and adiabatic reaction conditions. In particular, determine the reactor space times necessary to achieve 73% conversion for both modes of operation and the following parameter values for a (3/2)-order reaction (r = kc - ). 

So, for the same rate of ammonia removed from the gas (0.7052 moles per mole of air in), the differences between isothermal and adiabatic operation include (i) an order of magnitude smaller flow rate for water in the isothermal column and (ii) a factor of six larger mole fraction of ammonia at the liquid outlet from the isothermal column. The 42° difference in temperature causes a much greater solubility of ammonia in the liquid from the isothermal column however, a significant amount of heat must be removed to keep the temperature constant. This heat can be estimated from (12.4.26),... 

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. 

Numerical solutions to the coupled heat and mass balance equations have been obtained for both isothermal and adiabatic two- and three-transition systems but for more complex systems only equilibrium theory solutions have so far been obtained. In the application of equilibrium theory a considerable simplification becomes possible if axial dispersion is neglected and the plug flow assumption has therefore been widely adopted. Under plug flow conditions the differential mass and heat balance equations assume the hyperbolic form of the kinematic wave equations and solutions may be obtained in a straightforward manner by the method of characteristics. In a numerical simulation the inclusion of axial dispersion causes no real problem. Indeed, since axial dispersion tends to smooth the concentration profiles the numerical solution may become somewhat easier when the axial dispersion terra is included. Nevertheless, the great majority of numerical solutions obtained so far have assumed plug flow. 

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