Unsteady-state transport equation

In fact, since the unsteady-state transport equation for forced convection is linear, it is possible in principle to derive solutions for time-dependent boundary conditions, starting from the available step response solutions, by applying the superposition (Duhamel) theorem. If the applied current density varies with time as i(t), then the local surface concentration at any time c0(x, t) is given by... 

This problem has received special attention from electrochemists. If the rate of mass transfer is controlled by liquid diffusion, the governing unsteady-state transport equation... 

The turbulent transport equations are obtained in the traditional treatment of turbulence by time averaging the unsteady-state transport equations, after substituting the concentration and velocity components by the sum of their mean values and the corresponding time-dependent fluctuations. The following expression is thus obtained for the (time) average of the number of moles N that are transferred per unit time through unit area, in the direction y normal to the wall ... 

Let us close by noting that starting with the papers of Saltzman [117] and Lorentz [118], the problem of turbulent transport has been amenable to analysis on the basis of the unsteady-state transport equations without any additional assumptions. While this extremely meaningful procedure is outside the scope of the present contribution, at least a few [117-125] from a very large number of references need mention. 

There is also another key parameter linked to the choice of the material for the reactor. First, the choice is obviously determined by the reactive medium in terms of corrosion resistance. However, it also has an influence on the heat transfer abilities. In fact, the heat transport depends on the effusivity relative to the material, deflned by b = (XpCp) the effusivity b appears in the unsteady-state conduction equation. 

The transport of the adsorbed species into spherical particles is represented by the unsteady state diffusion equation as follows ... 

The driving potential assumed for moisture movement based on Equation 39 is the moisture concentration Other driving potentials may also be assumed. Table I lists the potentials that have been proposed, the resulting transport coefficients, and their relationships to D in each case (59). Although one or more of these other potentials may be more descriptive of the driving force for moisture movement, the discussion that follows will be restricted to the diffusion coefficient because it is so well established in the literature, and can be related to any of the others. Furthermore, it appears unchanged in the unsteady-state diffusion equation (Pick s second law), unlike any of the other coefficients. Thus Pick s second law may be written, for one dimension, as... 

The unsteady state balance equations must be completed with constitutive equations, which are relations between some state variables, usually expressing natural laws or the kinetics of transport phenomena. Examples are PVT relations, as equations of state, or kinetics expressions, phase-equilibrium factors, etc. Specifying the initial and boundary conditions completes the problem formulation. 

Transient moisture sorption under ramp changes in external humidity is analyzed. A general model describing the dynamics of moisture sorption is derived. The paper sheet is considered as a composite structure of fibers and voids through which moisture is transported by diffusion. The mathematical description of moisture transport embodies two suitably averaged concentration fields, c and q. Two unsteady state diffusion equations describe the time and spatial evolution of these fields. The average moisture content of the sheet and the moisture flux at the surface are evaluated. 

The model is composed by different equations which in all cases can be used in unsubscribed format in a basic language program. An important point to highlight is that Qwasi takes into account both steady and unsteady state solutions for the equations for systems involving contamination of lakes (or rivers). The equations considered by Qwasi involve more than 15 physicochemical processes (such as partitioning, sediment transport, deposition, etc.) to estimate the fate of the studied system. These processes and the main involved variables and parameters are summarized in Fig. 2. 

The basic equations for an unsteady-state process of one-dimensional (in the -direction) heat and mass transport with a simultaneous chemical reaction in a porous catalyst pellet are... 

The propeller setup was used for this purpose. From a computational standpoint, a mesh of the vessel-propeller set was created containing 8746 elements yielding 54,333 velocity equations and 8746 concentration equations. The surface mesh of the propeller (Fig. 5) comprised 964 control points. A maximum of three control points per element was used to avoid locking. Unsteady state flow simulations were performed with a 1-s time step and three coupling iterations between the Navier-Stokes equations and the solid transport equation were required per time step. Steady state was deemed obtained when the solids concentration coefficient of variation did not change. 

Distributed parameter, nonlinear, partial differential equations were soloed to describe oxygen transport from maternal to fetal bloody which flows in microscopic channels within the human placenta. Steady-state solutions were obtained to show the effects of variations in several physiologically important parameters. Results reported previously indicate that maternal contractions during labor are accompanied by a partially reduced or a possible total occlusion of maternal blood flow rate in some or all portions of the placenta. Using the mathematical modely an unsteady-state study analyzed the effect of a time-dependent maternal blood flow rate on placental oxygen transport during labor. Parameter studies included severity of contractions and periodicity of flow. The effects of axial diffusion on placental transport under the conditions of reduced maternal blood flow were investigated. 

Hence, the local mass transfer coefficient scales as the two-thirds power of a, mix for boundary layer theory adjacent to a solid-liquid interface, and the one-half power of A, mix for boundary layer theory adjacent to a gas-liquid interface, as well as unsteady state penetration theory without convective transport. By analogy, the local heat transfer coefficient follows the same scaling laws if one replaces a, mix in the previous equation by the thermal conductivity. 

Effective axial transport properties can be determined using an adiabatic reactor. Steady state mass and heat balances result in second-order ordinary differential equations when the axial dispersion is taken into consideration, solutions of which can readily be obtained. Based on these solutions and temperature or concentration measurements, the effective transport properties can be calculated in a manner similar to the procedures used for the radial transport properties. As indicated earlier, a transient experiment can also be used for the determination. Here, experimental and analytical procedures are illustrated for the determination of the effective axial transport property for mass. An unsteady state mass balance for an adiabatic reactor can be written as ... 

It must be stressed to the reader that the preceding discussion was based on the (simultaneous) steady state solution of the thermal and material transport equations. In a real physical situation the onset of ignition or extinction phenomena is inherently an unsteady state process it follows that the transition to ignition or extinction will take place at a finite rate. 

We will try our hand at applying the diffusion equation to a couple of mass transport problems. The first is the diffusive transport of oxygen into lake sediments and the use of oxygen by the bacteria to result in a steady-state oxygen concentration profile. The second is an unsteady solution of a spill into the groundwater table. 

---