Steady flows

In certain steady-flow systems, considerable simplification in equations (19)-(22) may be effected. Here d/dt = 0, the leading term on the left-hand side in equations (19)-(22) disappears, and d/dx d/dx because only one independent variable remains. Equation (19) may be integrated immediately, yielding the simplified continuity equation 

In many steady-state combustion problems, body forces are negligible and a consequent simplification in the energy equation is possible. When d/dt = 0 and = 0, multiplying equation (21a) by pgi9 and utilizing equation (23) shows that equation (21a) may be integrated once, giving 

Equation (24) states that the sum of the enthalpy flux and the ordered kinetic-energy flux may change only because of heat flux ( 2 3 ) viscous dissipation [ 2 PsiP Pi i) ] Equation (24) also illustrates that the enthalpy is of more importance than the internal energy in steady-flow problems the additional term p/p, which appears when equation (24) is written in terms of u, may be explained as displacement work.  

An integrated form of the momentum equation may be derived only under additional restrictive conditions. In steady-state flow with f = 0, if viscous forces are negligible, then equation (20) reduces to pv dv/dx - -dp/dx = 0. If, in addition, v /(p/p) 1, then d In p/dx d n v/dx (that is. 

Summary of Relevant Aspects of Fluid Dynamics and Chemical Kinetics 

The condition v f(p/p) 1 is valid for low-speed flow specifically, the square of the Mach number [.v /idpIdpX, s = entropy]—a measure of the ratio of the ordered kinetic energy to the random, thermal kinetic energy of the molecules—must be small. When equation (25) is an acceptable approximation, equations (22) (with dYJdt = 0) through (24) constitute an appropriate set of conservation equations. In these cases it is not necessary to use equation (20) (except to compute the actual small pressure change, after the rest of the problem has been solved with p = constant). 

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Open steady-flow systems, which include almost all air conditioning processes, foUow this law. Examples include the energy flows in a cooling and dehurnidifying coil or an evaporative cooling system. 

Hot fluid rises on one side and cold fluid fabs on the other side of the cell. The rising fluid cools as it nears the top, and the falling fluid warms as it nears the bottom, thus maintaining a steady flow. 

Automobiles are the largest source of obsolete scrap. Other important sources of obsolete scrap include the demolition of steel stmctures and railroad companies. The latter provide a steady flow of scrap from their fabricating shops and from the recovery of worn out or abandoned track and railroad cars. AH iron and steel products are recyclable if economically retrieved when scrapped. 

Energy Equations for Steady-State, Steady-Flow Processes... 

Fig. 4. Schematic diagram of a simple steady-flow process. See text.

Large quantities of fines in the furnace charge cause furnace blows that are hazardous to operator and equipment. Blows result when the steady flow of mix to part of the melt zone is stopped by a fines bridge the calcium carbide Hquid overheats and then dissociates at 2250°C. 

Real irreversible processes can be subjected to thermodynamic analysis. The goal is to calciilate the efficiency of energy use or production and to show how energy loss is apportioned among the steps of a process. The treatment here is limited to steady-state, steady-flow processes, because of their predominance in chemical technology. 

The energy balance for a steady-state steady-flow process resulting from the first law of thermodynamics is... 

ANALYSIS OF STEADY-STATE, STEADY-FLOW PROCESSES... 

FIG. 5-6 Temperature gradients for a steady flow of heat by conduction and convection from a warmer to a colder fluid separated by a solid wall. 

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

Here g is the gravity vector and tu is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integr on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors Vi and V9, the momentum equation becomes... 

Whenever relative motion exists between a particle and a surrounding fluid, the fluid will exert a drag upon the particle. In steady flow, the drag force on the particle is... 

Operation m the Simple Mode if there is no concentration gradient within the liquid pool and if there is no coalescence within the rising foam, then the operation shown by the sohd hnes of Fig. 22-42 is truly in the simple mode, i.e., a single theoretical stage of separation. Equations (22-45) and (22-46) will then apply to the steady-flow operation. 

Lapse Rate and Atmospheric Stability Apart from mechanical interference with the steady flow of air caused by buildings and other obstacles, the most important fac tor that influences the degree of turbulence and hence the speed of diffusion in the lower air is the varia-... 

Constant Flow into Protected Equipment For the steady-state design scenario with a constant, steady flow of fluid from a pressure source that is above the maximum aUowed pressure in the protected equipment, volume is being generated within the equipment at a rate RV = F/ f. Substituting into Eq. (26-21) and noting that the specific volume of the vent stream is l/p, gives the required mass flow rate ... 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

Equation 2.41 is the general energy equation for a steady flow process. 

A major cause of pulsing in flare systems is flow surging in the water seal drum. One of several reasons why it is important to eliminate pulsing is to reduce flare noise. Combustion flare noise has been shown to increase as the steam rate increases. Since the amount of steam required to suppress smoke in a flare is set by the flaring rate, flow surges will require a higher steam rate than for a steady flow. 

Other advantages of eliminating pulsing are reduced steam costs, more accurate flow measurements possible with a steady flow, and reduced incidence of blowing and seal liquid. Also, the size of the seal drum can sometimes be reduced. 

The effect of physical processes on reactor performance is more complex than for two-phase systems because both gas-liquid and liquid-solid interphase transport effects may be coupled with the intrinsic rate. The most common types of three-phase reactors are the slurry and trickle-bed reactors. These have found wide applications in the petroleum industry. A slurry reactor is a multi-phase flow reactor in which the reactant gas is bubbled through a solution containing solid catalyst particles. The reactor may operate continuously as a steady flow system with respect to both gas and liquid phases. Alternatively, a fixed charge of liquid is initially added to the stirred vessel, and the gas is continuously added such that the reactor is batch with respect to the liquid phase. This method is used in some hydrogenation reactions such as hydrogenation of oils in a slurry of nickel catalyst particles. Figure 4-15 shows a slurry-type reactor used for polymerization of ethylene in a sluiTy of solid catalyst particles in a solvent of cyclohexane. 

It is possible to determine the value of N that gives the best fit to the response eurve of the aetual reaetor as deseribed below. Consider a steady flow u m /see of fluid in and out of the first reaetor volume... 

Consider a steady flow of reaetant A to produets at eonstant density through an element of radius r, width 6r, and height 61 in a tubular reaetor at isothermal eondition. Suppose that radial and axial mass transfer is expressed by Fiek s law, with (Dg)[ and (Dg) as effeetive diffusivities. The rate at whieh A reaets is (-i ), mol/m see. A material balanee on a tubular element of radii r and r -i- 6r and height 61 is earried out from... 

For steady flow there must be equilibrium of forces so... 

Solution The steady flow energy equation may be written as... 

Glasser, D., Hildebrandt, D., and Crowe, C. (1987). A geometric approach to steady flow reactors The attainable region and optimization in concentration space. Ind. Eng. Chem. Res., 26, 1803-1810. 

A flow is uniform when the velocity of flow is the same at any given instant at every point in the fluid. This state of affairs can exist only with an ideal fluid. However, steady flow (uniform flow) is assumed to take place in a duct with the velocity constant along a streamline. 

A streamline shows the direction of a number of particles of fluid at the same instant in time. Flow cannot take place across a streamline. Path lines and streamlines will be identical for steady flow. 

The discussion here is restricted to plants in which the flow is steady, since virtually all the plants (and their components) with which the book is concerned have a steady flow. 

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