Constant flux boundary condition

The above self-similar velocity profiles exists only for a Re number smaller than a critical value (e.g. 4.6 for a circular pipe). The self-similar velocity profiles must be found from the solution of the Navier-Stokes equations. Then they have to be substituted in Eq. (25) which must be solved to compute the local Nusselt number Nu z). The asymptotic Nusselt number 7Vm is for a pipe flow and constant temperature boundary condition is given by Kinney (1968) as a function of Rew and Prandtl (Pr) numbers. The complete Nu(z) curve for the pipe and slit geometries and constant temperature or constant flux boundary conditions were given by Raithby (1971). This author gave /Vm is as a function of Rew and fluid thermal Peclet (PeT) number. Both authors solved Eq. (25) via an eigenfunction expansion. 

Laplace transformation to the constant-flux boundary condition (4.50). Laplace transformation on the left-hand side of the boundary condition leads to (dc /dx), and the same operation performed on the right-hand side, to - l/Dp (Appendix 4.2). Thus, from the boundary condition (4.50) one gets... 

Asymptotic value for the Nusselt number for heat transfer with a constant flux boundary condition 2. 

The source term can be included as a constant flux boundary condition at the lowest wavenumber, ks = 2n/Ls The terms on the right represent the decay of fluctuations due to (i) the reaction acting uniformly on all scales, and (ii) due to diffusion, that becomes stronger at large wavenumbers. For large Pe the diffusive decay is relatively weak in the inertial convective range and can be neglected for simplicity. 

Step 11. Write all the boundary conditions that are required to solve this boundary layer problem. It is important to remember that the rate of reactant transport by concentration difhision toward the catalytic surface is balanced by the rate of disappearance of A via first-order irreversible chemical kinetics (i.e., ksCpJ, where is the reaction velocity constant for the heterogeneous surface-catalyzed reaction. At very small distances from the inlet, the concentration of A is not very different from Cao at z = 0. If the mass transfer equation were written in terms of Ca, then the solution is trivial if the boundary conditions state that the molar density of reactant A is Cao at the inlet, the wall, and far from the wall if z is not too large. However, when the mass transfer equation is written in terms of Jas, the boundary condition at the catalytic surface can be characterized by constant flux at = 0 instead of, simply, constant composition. Furthermore, the constant flux boundary condition at the catalytic surface for small z is different from the values of Jas at the reactor inlet, and far from the wall. Hence, it is advantageous to rewrite the mass transfer equation in terms of diffusional flux away from the catalytic surface, Jas. 

The constant flux boundary condition implies that sufficient ions are delivered to the root surface so that uptake is controlled only by plant demand (De Willigen and Van Noordwijk, 1994b) ... 

An additional test for diffusion control is provided by the dependence of the steady-state anodic current on membrane thickness. In potentiostatic charging, the hydrogen concentration is fixed in the case of pure diffusion control, and therefore should be proportional to 1/L (see Section III). This test is not necessarily applicable for a constant flux boundary condition because C /L,... 

In general, the axial heat conduction in the channel wall, for conventional size channels, can be neglected because the wall is usually very thin compared to the diameter. Shah and London (1978) found that the Nusselt number for developed laminar flow in a circular tube fell between 4.36 and 3.66, corresponding to values for constant heat flux and constant temperature boundary conditions, respectively. 

The contaminant transport model, Eq. (28), was solved using the backwards in time alternating direction implicit (ADI) finite difference scheme subject to a zero dispersive flux boundary condition applied to all outer boundaries of the numerical domain with the exception of the NAPL-water interface where concentrations were kept constant at the 1,1,2-TCA solubility limit Cs. The ground-water model, Eq. (31), was solved using an implicit finite difference scheme subject to constant head boundaries on the left and right of the numerical domain, and no-flux boundary conditions for the top and bottom boundaries, corresponding to the confining layer and impermeable bedrock, respectively, as... 

Using Pick s law, Ihe constant species flux boundary condition for a diffusing species A at a boundary at. v = 0 is expressed, in the absence of any blowing or suction, as... 

Whereas the circumferential variations of the local wall shear stress (i.e., the momentum flux) in itself are not of interest in the study of the BSR, the analogous variations in mass flux or surface concentration are indeed. In Ref. 15 a graph is presented of the local heat flux relative to the circumferential average, for the constant-temperature boundary condition, as a function of a and s/dp. These data are based on a semianalytical solution of the governing PDE, following the procedure described by Ref. 8 (see Section II.B.2). At a relative pitch of 1.2 the local flux at a = 0 is ca. 64% lower than the circumferential average at a relative pitch of 1.5 the flux at a = 0 is still ca. 20% lower than the circumferential average. In the case of a constant surface temperature, the local heat fluxes are directly proportional to the local Nusselt (or Sherwood) numbers. 

Table 2 demonstrates the effects of the Knudsen and the Brinkman numbers on heat transfer in a tube flow. As it can be seen, the Nusselt number decreases with the increases in both the Brinkman number and the Knudsen number, since the increasing temperature jump decreases heat transfer. Also, under the constant wall temperature boundary conditions, the Nusselt numbers are greater than under constant heat flux boundary conditions when the Brinkman number is nonzero [51, 521. 

Hydrodynamically fully-developed laminar gaseous flow in a cylindrical microchannel with constant heat flux boundary condition was considered by Ameel et al. [2[. In this work, two simplifications were adopted reducing the applicability of the results. First, the temperature jump boundary condition was actually not directly implemented in these solutions. Second, both the thermal accommodation coefficient and the momentum accommodation coefficient were assumed to be unity. This second assumption, while reasonable for most fluid-solid combinations, produces a solution limited to a specified set of fluid-solid conditions. The fluid was assumed to be incompressible with constant thermophysical properties, the flow was steady and two-dimensional, and viscous heating was not included in the analysis. They used the results from a previous study of the same problem with uniform temperature at the boundary by Barron et al. [6[. Discontinuities in both velocity and temperature at the wall were considered. The fully developed Nusselt number relation was given by... 

The laminar gaseous flow heat convection problem was solved in a cylindrical microchannel with uniform heat flux boundary conditions in [20]. The fluid was assumed to be incompressible with constant properties, the flow was assumed to be steady and two-dimensional, and viscous heating was neglected. They used the results from a previous study, [21], of the same problem with uniform... 

The solution for this equation is the same as that of the constant potential boundary conditions (Dirichlet s problem) and was solved not only for electrostatic fields but also for heat fluxes and concentration gradients (chemical potentials) [10]. The primary potential distribution between two infinitely parallel electrodes is simply obtained by a double integration of the Laplace equation 13.5 with constant potential boundary conditions (see Figure 13.3). The solution gives the potential field in the electrolyte solution and considering that the current and the electric potential are orthogonal, the direct evaluation of one function from the other is obtained from Equation 13.7. 

In this work, heat and fluid flow in some common micro geometries is analyzed analytically. At first, forced convection is examined for three different geometries microtube, microchannel between two parallel plates and microannulus between two concentric cylinders. Constant wall heat flux boundary condition is assumed. Then mixed convection in a vertical parallel-plate microchannel with symmetric wall heat fluxes is investigated. Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar flow having constant properties (i.e. the thermal conductivity and the thermal diffusivity of the fluid are considered to be independent of temperature) is considered. The axial heat conduction in the fluid and in the wall is assumed to be negligible. In this study, the usual continuum approach is coupled with the two main characteristics of the microscale phenomena, the velocity slip and the temperature jump. 

In this lecture, the effects of the abovementioned dimensionless parameters, namely, Knudsen, Peclet, and Brinkman numbers representing rarefaction, axial conduction, and viscous dissipation, respectively, will be analyzed on forced convection heat transfer in microchannel gaseous slip flow under constant wall temperature and constant wall heat flux boundary conditions. Nusselt number will be used as the dimensionless convection heat transfer coefficient. A majority of the results will be presented as the variation of Nusselt number along the channel for various Kn, Pe, and Br values. The lecture is divided into three major sections for convective heat transfer in microscale slip flow. First, the principal results for microtubes will be presented. Then, the effect of roughness on the microchannel wall on heat transfer will be explained. Finally, the variation of the thermophysical properties of the fluid will be considered. 

Nonisotfaermal reactor with isothermal cooling jacket. A coolant at constant temperature cooling jacket is added to the previous example to examine the perfomiancc of a nonisothermal reactor. In thi.s model, the boundary condition for the energy balance at the radial boundary is changed from the thermal insulation boundary condition to a heat flux boundary condition. 

A common Alternative boundary condition to that of constant concentration at the outer surface is the flux boundary condition in which the (lux of material on of the slab is taken as proportional to the difference in the concentration at the outer surface of the slab and that far into the surrounding medium ... 

Thermally Developing Flow. Numerous investigators [80, 89-94] have carried out the investigation of turbulent thermally developing flow in a smooth circular duct with uniform wall temperature and uniform wall heat flux boundary conditions. It has been found that the dimensionless temperature and the Nusselt number for thermally developing turbulent flow have the same formats as those for laminar thermally developing flow (i.e., Eqs. 5.34-5.37 and Eqs. 5.50-5.53). The only differences are the eigenvalues and constants in the equations. 

The fully established laminar heat transfer results for nonnewtonian fluids flowing through a circular tube with a fully developed velocity distribution and constant heat flux boundary condition at the wall can be obtained by solving the following energy equation ... 

The Nusselt number for power-law fluids for constant wall heat flux reduces to the newto-nian value of 4.36 when n = 1 and to 8.0 when n = 0. Equation 10.47 is applicable to the laminar flow of nonnewtonian fluids, both purely viscous and viscoelastic, for the constant wall heat flux boundary condition for values of xId beyond the thermal entrance region. The laminar heat transfer results for the constant wall temperature boundary condition were also obtained by the separation of variables using the fully developed velocity profile. The values of the Nusselt number for n = 1.0, Vi, and A calculated by Lyche and Bird [40] are 3.66, 3.95, and 4.18, respectively, while the value for n = 0 is 5.80. These values are equally valid for purely viscous and viscoelastic fluids for the constant wall temperature case provided that the thermal conditions are fully established. 

It is interesting to note that the nonnewtonian effect has been taken into account by simply multiplying the corresponding newtonian result by [(3n + l)/4n]1/3. Equations 10.48 and 10.49 may be used to predict the local heat transfer coefficient of purely viscous and viscoelastic fluids in the thermal entrance region of a circular tube. Figure 10.6 shows a typical comparison of the measured local heat transfer coefficient of a viscoelastic fluid with the prediction for a power-law fluid. The good agreement provides evidence to support the applicability of Eq. 10.48 in the case of the constant heat flux boundary condition. 

FIGURE 10.6 Experimental results for laminar pipe flow heat transfer for constant wall heat flux boundary conditions [35]. 

Local heat transfer measurements were carried out in the once-through system for the same aqueous polyacrylamide solutions used in the friction factor and viscosity measurements shown in Figs. 10.22 and 10.23 [37, 93]. These heat transfer studies involving a constant heat flux boundary condition required the measurement of the fluid inlet and outlet temperatures and the local wall temperature along the tube. These wall temperatures are presented in terms of a dimensionless wall temperature 0 in Fig. 10.27 for four selected concentrations. Here 0 is defined as... 

Ra, Rayleigh number for constant heat flux boundary condition, Gr, Pr Re+ Reynolds number = p U2 "dR/K... 

If a constant heat flux boundary condition is required, an electrical heating element, often a thin, metallic foil, can be stretched over an insulated wall. The uniform heat flux is obtained by Joule heating. If the wall is well insulated, then, under steady-state conditions, all of the energy input to the foil goes to the fluid flowing over the wall. Thermocouples attached to the wall beneath the heater can be used to measure local surface temperature. From the energy dissipation per unit time and area, the local surface temperature, and the fluid temperature, the convective heat transfer coefficient can be determined. Corrections to the total heat flow (e.g., due to radiation heat transfer or wall conduction) may have to be made. 

H Constant axial wall heat flux boundary condition... 

Such a boundary condition is an hypothetical state for the materials, and does not exactly reflect reality for at least two reasons. The EB is allowed to move in direction of the canister, but no fluids fluxes are allowed. If there were a liquid flow in the void, it would probably vaporize, because of gas circulation in the neighbourhood of the chamber. The flux boundary condition would neither be a null flux, nor be a constant capillary pressure. Moreover, the dependence between capillary pressure and time at this point would be very hard to determine. We then assume that, if there is a fluid flux in the void, it would not be important and can be neglected. 

Unfortunately, for a given domain of interest boundary conditions can be chosen that over- or underspecify the problem. An example of an over-specified problem is a constant area duct, with a fixed fluid velocity at the inlet and but different at the outlet. Naturally, for an incompressible fluid both conditions cannot be physically satisfied in the absence of a mass source. In the same sense, a closed box with only heat flux boundary conditions is under specified since the temperature level is not constrained, and therefore unpredictable. It becomes clear that defining well posed boundary conditions is quite important for the proper solution of a flow problem. An easy way to check for well posed boundary conditions is to ask yourself Could the chosen configuration be physically recreated in the laboratory . 

Since there was no current flow in the refractory walls, the magnetic vector potential. A, was set equal to zero. For heat transfer, a constant heat flux boundary condition, equal to the measured heat loss flux, was specified for this wall. The heat loss fluxes at the side wall for the water-cooled copper panels in Ae bullion and slag were 31.3 kW/m and 1.75 kW/m respectively. The conventional non-slip boundary condition was used for momentum transfer. 

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