Convection example

Convection. When a current or macroscopic particle of fluid crosses a specific surface, such as the boundary of a control volume, it carries with it a definite quantity of enthalpy. Such a flow of enthalpy is called a convective flow of heat or simply convection. Since convection is a macroscopic phenomenon, it can occur only when forces act on the particle or stream of fluid and maintain its motion against the forces of friction. Convection is closely associated with fluid mechanics. In fact, thermodynamically, convection is not considered as heat flow but as flux of enthalpy. The identification of convection with heat flow is a matter of convenience, because in practice it is difficult to separate convection from true conduction when both are lumped together under the name convection. Examples of convection are the transfer of enthalpy by the eddies of turbulent flow and by the current of warm air from a household furnace flowing across a room. 

It is also essential that the period of the ac stimulus not be so long that convection becomes a factor within a few cycles. The lower frequency limit was set here at 1 Hz because convection would become a problem in the range of several seconds in most liquid systems with water-like viscosity. Current equipment for EIS can operate at much lower frequencies (as low as 10 jU,Hz) and can be usefully applied in the low-frequency (long-time) regime when the processes being examined are not controlled by convection. Examples include transport or reaction at a solid-solid interfaces or diffusion and reaction in extremely viscous media, such as glasses or polymers. 

Figure 2.41 Velocity contours due to thermal convection Example 2.11 n = 1, Eq. (2.106)).

The results from this thermal-convection example and the results from the natural convection-diffusion problem presented in the last section of this chapter have a basic difference the vertical velocity. For the former, the vertical velocity in the whole distance across the reservoir is effective. For the latter, it is significantly close to the side boundaries and is close to zero across the major part of the reservoir (compare Figs. 2.30 and 2.43). 

The Maxwell class of viscoelastic constitutive equations are described by a simpler form of Equation (1.22) in which A = 0. For example, the upper-convected Maxwell model (UCM) is expressed as... 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

Keeping all of the flow regime conditions identical to the previous example, we now consider a finite element model based on treating silicon rubber as a viscoelastic fluid whose constitutive behaviour is defined by the following upper-convected Maxwell equation... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

Convective heat transfer is classified as forced convection and natural (or free) convection. The former results from the forced flow of fluid caused by an external means such as a pump, fan, blower, agitator, mixer, etc. In the natural convection, flow is caused by density difference resulting from a temperature gradient within the fluid. An example of the principle of natural convection is illustrated by a heated vertical plate in quiescent air. 

Effect of Uncertainties in Thermal Design Parameters. The parameters that are used ia the basic siting calculations of a heat exchanger iaclude heat-transfer coefficients tube dimensions, eg, tube diameter and wall thickness and physical properties, eg, thermal conductivity, density, viscosity, and specific heat. Nominal or mean values of these parameters are used ia the basic siting calculations. In reaUty, there are uncertainties ia these nominal values. For example, heat-transfer correlations from which one computes convective heat-transfer coefficients have data spreads around the mean values. Because heat-transfer tubes caimot be produced ia precise dimensions, tube wall thickness varies over a range of the mean value. In addition, the thermal conductivity of tube wall material cannot be measured exactiy, a dding to the uncertainty ia the design and performance calculations. 

The values of CJs are experimentally determined for all uncertain parameters. The larger the value of O, the larger the data spread, and the greater the level of uncertainty. This effect of data spread must be incorporated into the design of a heat exchanger. For example, consider the convective heat-transfer coefficient, where the probabiUty of the tme value of h falling below the mean value h is of concern. Or consider the effect of tube wall thickness, /, where a value of /greater than the mean value /is of concern. 

The effective thermal conductivity of a Hquid—soHd suspension has been reported to be (46) larger than that of a pure Hquid. The phenomenon was attributed to the microconvection around soHd particles, resulting in an increased convective heat-transfer coefficient. For example, a 30-fold increase in the effective thermal conductivity and a 10-fold increase in the heat-transfer coefficient were predicted for a 30% suspension of 1-mm particles in a 10-mm diameter pipe at an average velocity of 10 m/s (45). 

The heat pipe has properties of iaterest to equipmeat desigaers. Oae is the teadeacy to assume a aeady isothermal coaditioa while carrying useful quantities of thermal power. A typical heat pipe may require as Htfle as one thousandth the temperature differential needed by a copper rod to transfer a given amount of power between two poiats. Eor example, whea a heat pipe and a copper rod of the same diameter and length are heated to the same iaput temperature (ca 750°C) and allowed to dissipate the power ia the air by radiatioa and natural convection, the temperature differential along the rod is 27°C and the power flow is 75 W. The heat pipe temperature differential was less than 1°C the power was 300 W. That is, the ratio of effective thermal conductance is ca 1200 1. 

Under conditions of limiting current, the system can be analyzed using the traditional convective-diffusion equations. For example, the correlation for flow between two flat plates is... 

Tank Cells. A direct extension of laboratory beaker cells is represented in the use of plate electrodes immersed into a lined, rectangular tank, which may be fitted with a cover for gas collection or vapor control. The tank cell, which is usually undivided, is used in batch or semibatch operations. The tank cell has the attraction of being both simple to design and usually inexpensive. However, it is not the most suitable for large-scale operation or where forced convection is needed. Rotating cylinders or rotating disks have been used to overcome mass-transfer problems in tank cells. An example for electroorganic synthesis is available (46). 

Example Consider the equation for convection, diffusion, and reaction in a tiihiilar reactor. 

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