Volume-averaged enthalpy

For simplicity, consider an incompressible medium flowing through the multichannel domain depicted in Figure 2.36. In a number of practical applications the heat flux in the x-direction will be very small compared with that in the y-direction. Then the volume-averaged enthalpy equation for the solid walls can be written as... 

Figure 2.36 Multichannel flow domain with typical averaging volume for obtaining volume-averaged enthalpy equations.

The modified volume averaged enthalpy equation can be deduced from (3.172) or alternatively from (3.195) and the relations between the averaged internal energy and the enthalpy ... 

For incompressible and porous media flows we may follow the approaches of [239, 200] wherein the variations of the factor PkCp k within the averaging volume are neglected to enable application of the Leibnitz and Gauss averaging rules on the temperature and apparent flux terms. For this particular case the modified volume averaged enthalpy equation in terms of temperature can be deduced from (3.176) as follows ... 

The volume averaged enthalpy equation is given by [205, 219] - iaic pkhk Vk ) + oik pkykhk)Vk)... 

Predictions of the vapor volume and enthalpy departure for a water containing natural gases were compared with data being generated by Hall and coworkers at Texas A M under GPA sponsorship (10). Both equations of state performed well the average error Tn the predicted volume was less than one percent (abs) and the absolute average error in the predicted enthalpy departure was about four KJ/kg. 

Fuel pellet Temperatnre <2800°C Volume averaged fnel enthalpy 10 kJ/kg... 

Specific volume v J, thermal expansion coefficient, heat capacity at constant pressure (cj, heat capacity at constant volume (cj, enthalpy, entropy, and thermal conductivity (X) are thermal characteristics relevant for processing and applications. They depend on PE grade, temperature, pressure, average molecular weight, branching, crystallinity or density, stretching ratio, heating rate, spherulite size, and so on. 

Where ( ) is the internal microwave power dissipated per unit volume, K[/w ] is permeability, and/z is the averaged enthalpy. In Eq. 16... 

Fig. 3.1 a Tempraatme dependence of volume or enthalpy for an amtnphous polymer. The vertical lines denote Tg determined using fast and slow cooling rates. The vatical arrow denotes glassy-state structural relaxation (physical aging) of the glass ftnmed on slow cooling, b Normalized Arrhenius plot of the average alpha-relaxation time fra- an amorphous polymer. Above Tg, polymer relaxation exhibits a non-Anhenius tempaature dependence while below Tg, polymer relaxation exhibits an Arrhenius temperature dependence... 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

The standard enthalpy change for this reaction at 200 °C is —126 kJ/mole for the reaction as written. Each CSTR has a volume of 0.2 m3. The feed rate is 66 cm3/sec. The following data on average heat capacities are available. 

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