Assumption isothermal

Mak s article show s how the isothermal assumption is slightly conservative (higher relief valve outlet pressures) when compared to adiabatic. 

To rationalize the isothermal assumption, Dykhuizen 39() discussed two related physical phenomena. First, heat may be drawn out of the substrate from an area that is much larger than that covered by asplat. Thus, the 1 -D assumption in the Stefan problem becomes invalid, and a solution of multidimensional heat conduction may make the interface between a splat and substrate closer to isothermal. Second, the contact resistance at the interface is deemed to be the largest thermal resistance retarding heat removal from the splat. If this resistance does not vary much with substrate material, splat solidification should be independent of substrate thermal properties. Either of the phenomena would result in a heat-transfer rate that is less dependent on the substrate properties, but not as high as that calculated by Madej ski based on the... 

The speed of the adsorption wave can be readily derived by introducing the linear isotherm assumption and the chain mle derivative of q with respect to t. The wave speed results because the assumptions turn Eq. (9.10) into a kinematic wave equation and the wave speed W is instantly recognized as ... 

Parallel-Plate Flow of Newtonian Fluids A Newtonian polymeric melt with viscosity 0.21b(S/in2 and density 481b/ft3, is pumped in a parallel-plate pump at steady state and isothermal conditions. The plates are 2 in wide, 20 in long, and 0.2 in apart. It is required to maintain a flow rate of 50 lb/h. (a) Calculate the velocity of the moving plate for a total pressure rise of 100 psi. (b) Calculate the optimum gap size for the maximum pressure rise, (c) Evaluate the power input for the parts (a) and (b). (d) What can you say about the isothermal assumption ... 

Calculating the adiabatic temperature rise can make an approximate test of the validity of the isothermal assumption is obtained from Eq. E9.1-9... 

If AT is significant, we reject the isothermal assumption, but if it is small, local temperature effects may still be important.6... 

The fully filled channel and the isothermal assumptions are not realistic in that, in practice, channels are partially filled and the flow is nonisothermal. The constitutive equation and the equations of change used are ... 

Experiments were first performed using sieve pellets containing an imbedded thermocouple and with gas concentrations up to 10% ethane. These showed temperature variations of less than 0.5 °C during sorption, thus confirming the isothermal assumption. 

The need to govern heat balances properly in membrane reactors will certainly become a major task if large-scale industrial units are ever to be put into operation. Whether the performed reaction is endothermic (dehydrogenation) or exothermic (oxidation), innovative means to supply or remove heat from large-scale membrane reactor modules will have to be designed. The isothermicity assumption valid for several lab-scale membrane reactors will not hold anymore, and much more complex modeling will certainly have to be developed. 

For the conditions of Prob. 10-2 predict the temperature difference between bulk gas and pellet surface at 350 and at 500°C. Comment on the validity of the isothermal assumptions made in Probs. 10-1 to 10-3. 

A further issue concerns the possible intrusion of heat effects. This issue has been addressed both theoretically and experimentally [52,57], and a simple criterion for the vaUdity of the isothermal approximation was developed. The isothermal assumption is normally vahd for measurements with zeoHte crystals (< 150 pm), but for large macroporous particles some intrusion of heat-transfer effects can be anticipated when the heat of adsorption is large. 

In the earth atmosphere, there are usually certain fluxes of heat, therefore both approaches that treat the gas column as a closed thermodynamic system are not correct. The adiabatic case is more suitable, if the respective process of movement of a gas to some height is fast in comparison to the rate of heat conduction, thus entropy exchange of the gas. It was pointed out by Schaefer that the isothermal assumption in the barometric formula is highly unrealistic, and the adiabatic equation should be used [13]. 

This is done in Fig. 8.1 in the form of adsorption uptake curves. For the range Titr,<0.1 the isothermal assumption is reasonable, while uptake curves deviate from q. (8-13) for larger Thr.. 

For the sake of simplicity many fixed bed designs incorporate the isothermal assumption which is normally valid when the adsorbable component concentration is low, or the heat of adsorption is low, or the thermal wave and the mass transfer zone are well separated at the end of the bed. A relatively simple method is available to test whether the last criterion is valid (Ruthven 1984). 

When R = 1 the thermal wave progresses through the bed more or less at the same speed as the mass transfer zone. Hence, virtually all the heat released on adsorption can be expected to be retained in the MTZ and the isothermal assumption should not be made unless either the heat of adsorption is low and/or the concentration of the adsorbable component is low. When R is very much less than unity the thermal wave lags behind the MTZ and hence the heat of adsorption can be retained in the equilibrium portion of the bed (that is, from the entrance up to Le shown in Figure 5.6 (b)). Retention of the heat of adsorption in this way is beneficial to the subsequent desorption step (Garg and Ausikaitis 1983). When R is very much greater than unity the heat is easily removed from the MTZ and it is safe to invoke the isothermal assumption. Further discussion on the crossover ratio is given in Section 7.5.3. 

For fundamental research where the thermal parameters are not precisely known and to define unknown thermal transport parameters, we would like to directly measure the temperature profile within the electrolyte. To approach this problem, a thermocouple can be embedded directly in the diffusion media of a PEFC [32, 33]. However, the contact resistance between the diffusion media and the thermocouple becomes another unknown parameter. To circumvent these difficulties, Burford et al. invented a method to embed an array of microthermocouples directly between two 25-pm-thick Nafion electrolyte sheets of a membrane electrode assembly [34, 35]. Local temperature variation in PEFCs was determined to reach > 10°C at high current density for a thick diffusion media (>400 pm for woven cloth media). This proved that an isothermal assumption is typically not justified over a full range of performance and indicates phase change plays a role in water transport in PEFCs. An even smaller MEMs-based thermosensor array has been developed using vapor deposition [36] and has been embedded within a PEFC electrolyte, providing precise locational control of the sensor position. 

Recently Rao, Berne, and Kalos have reported Monte Carlo and molecular dynamics studies of the homogeneous nucleation of a Lennard-Jones fluid (argon). Several features of this study are noteworthy. Perhaps most importantly, the authors conclude from their dynamical study that growth of the nucleation microclusters takes place under essentially adiabatic conditions. That is, collisions between the microcluster and the vapor are not sufficiently frequent to maintain the droplet at a constant temperature. This result suggests that the isothermal assumption common in conventional nucleation theories... 

Figure 7. Maximum viscous deviation compared to the isothermal assumption, as a function of the pressure difference.

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