Flow isothermal

At very low densities It Is quite easy Co give a theoretical description of thermal transpiration, alnce the classical theory of Knudsen screaming 9] can be extended to account for Che Influence of temperature gradients. For Isothermal flow through a straight capillary of circular cross-section, a well known calculation [9] gives the molar flux per unit cross-sectional area, N, In the form... 

As already mentioned, the present code corresponds to the solution of steady-state non-isothennal Navier-Stokes equations in two-dimensional Cartesian domains by the continuous penalty method. As an example, we consider modifications required to extend the program to the solution of creeping (Stokes) non-isothermal flow in axisymmetric domains ... 

Steady-state, laminar, isothermal flow is assumed. For a given viscometer with similar fluids and a constant pressure drop, the equation reduces to 77 = Kt or, more commonly, v = r /p = Ct where p is the density, V the kinematic viscosity, and C a constant. Therefore, viscosity can be determined by multiplying the efflux time by a suitable constant. 

For banks of staggered tubes, the friction factor for isothermal flow is obtained from Fig. (6-42). Each fence (group of parametric curves) represents a particular Reynolds number defined as... 

For banks of in-line tubes,/for isothermal flow is obtained from Fig. 6-43. Average deviation from available data is on the order of 15 percent. For tube spacings greater than 3D(, the charts of Gram, Mackey, and Monroe (Trans. ASME, 80, 25—35 [1958]) can be used. As an approximation, the pressure drop can be taken as 0.32 velocity head (based on V ) per row of tubes (Lapple, et al.. Fluid and Paiiicle Mechanics, University of Delaware, Newark, 1954). 

A quick method for sizing compressible isothermal flow is offered by the following method developed by Lapple. 

This method employs a theoretical critical mass flow based on an ideal nozzle and isothermal flow condition. For a pure gas, the mass flow can be determined from one equation ... 

It will be shown in Chapter 5 that the pressure drop, AP, for isothermal flow in a circular section channel is given by... 

The steady isothermal flow of incompresible fluids through straight horizontal tubes is of importance in a number of cases of practical interest. 

Derive expressions for the velocity profile, shear stress, shear rate and volume flow rale during the isothermal flow of a power law fluid in a rectangular section slit of width W, depth H and length L. During tests on such a section the following data was obtained. 

A polymer melt is injected into a circular section channel under constant pressure. What is the ratio of the maximum non-isothermal flow length to the isothermal flow length in the same time for (a) a Newtonian melt and (b) a power law melt with index, n = 0.3. 

Miillejans has reported that with air supply through rectangular openings the jet behaves more or less as an isothermal flow when Ar < 101... 

Nielsen, P.V., L. Evenson, P. Grabau, and J.H. Thulesen-Dahl. 1987. Air distribution in rooms with ceiling-mounted obstacles and three-dimensional isothermal flow. RoomVent 87. Proceedings of the International Conference on Air Distribution in Ventilated Spaces, Stockholm. 

Scaling laws If a full-scale test is not possible, reduced-scale experiments are a good alternative. However, certain scaling laws must be observed (see Section 12.4, Scale model experiments ). Correct scaling for isothermal flows is usually possible. However, scaling of buoyant flows in large rooms may be difficult, if not impossible. Then numerical simulation is the better choice. 

For the isothermal flow of an ideal gas, then from equation 2.69 ... 

It may be noted that equations 4.29, 4.30 and 4.31 give a linear relation between Gmax and Pi. Comparison with equation 4.15 shows that the maximum flowrate Gmax is (0.685/0.607) = 1.13 times greater than for isothermal flow for a diatomic gas. 

In considering the flow in a pipe, the differential form of the general energy balance equation 2.54 are used, and the friction term 8F will be written in terms of the energy dissipated per unit mass of fluid for flow through a length d/ of pipe. In the first instance, isothermal flow of an ideal gas is considered and the flowrate is expressed as a function of upstream and downstream pressures. Non-isothermal and adiabatic flow are discussed later. 

Figure 4.10. Mass flowrate as function of downstream pressure for isothermal flow of air in pipe (l = 14 m.

Little error is introduced if this expression is applied to the flow of a compressible fluid provided that the velocity is not greater than about 60 m/s. When the velocity is high, the equation of state must be used to give the relation between the pressure and the volume of the gas. For non-isothermal flow, Pvk = a constant,... 

For an ideal gas in non-isothermal flow. If the pressure and volume are related bv Pi — constant, then a similar analysis gives ... 

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