Fluid isothermal pipe flow

Figure 3.18 Radial profile of shear stress, shear rate, and speed through the cross-section for an isothermal pipe flow with Newtonian, shear thinning, and shear thickening fluids...

Show that the volumetric flowrate of this fluid in a horizontal pipe of radius a under isothermal laminar flow conditions with a pressure gradient —AP/l per unit length is ... 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian liquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic fluid described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

Fig. 9.11 The RTD function F(t) versus reduced time t/t for flow in screw extruder compared to plug flow, isothermal flow of Newtonian fluids in pipes, and a continuously stirred tank vessel (CST).

Table 3.2 Summary of the relations for Newtonian and shear thinning/shear thickening fluids in the case of a simple fully developed pipe flow (isothermal) [61, [14]...

The temperature of the fluid in compressible flow through a conduit of constant cross section may be kept constant by a transfer of heat through the conduit wall. Long, small, uninsulated pipes in contact with air transmit suflicient heat to keep the flow nearly isothermal. Also, for small Mach numbers, the pressure pattern for isothermal flow is nearly the same as that for adiabatic flow for the same entrance conditions, and the simpler equations for isothermal flow may be used. The maximum velocity attainable in isothermal flow is... 

This is the equation for the velocity of sound in the fluid at the conditions for isothermal flow. Thus, for isothermal compressible flow there is a maximum flow for a given upstream p, and further reduction of p2 will not give any further increase in flow. Further details as to the length of pipe and the pressure at the maximum flow conditions are discussed elsewhere (Dl M2, PI). 

Equation 7.442 is a very useful relationship for determining fully developed temperature profiles in pipe flow law fluids. The maximum fully developed temperature always occurs at the center of the flow channel as can be seen from Eq. 7.442 as well as from Fig. 7.111, which illustrates the fully developed temperature profile under isothermal wall conditions and at various values of the power law index. 

Problem 10-8 (Level 2) A Newtonian fluid is flowing in fully developed, isothermal, laminar flow through a tube of radius R and length L. The volumetric flow rate is v. The velocity distribution in the pipe is... 

Compressible fluid flow occurs between the two extremes of isothermal and adiabatic conditions. For adiabatic flow the temperature decreases (normally) for decreases in pressure, and the condition is represented by p V (k) = constant. Adiabatic flow is often assumed in short and well-insulated pipe, supporting the assumption that no heat is transferred to or from the pipe contents, except for the small heat generated by fricdon during flow. Isothermal pVa = constant temperature, and is the mechanism usually (not always) assumed for most process piping design. This is in reality close to actual conditions for many process and utility service applications. 

Compressibility of a gas flowing in a pipe can have significant effect on the relation between flowrate and the pressures at the two ends. Changes in fluid density can arise as a result of changes in either temperature or pressure, or in both, and the flow will be affected by the rate of heat transfer between the pipe and the surroundings. Two limiting cases of particular interest are for isothermal and adiabatic conditions. 

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. 

It has been shown in equation 4.35 that this velocity vw is equal to the velocity of transmission of a small pressure wave in the fluid at the pressure Pw if heat could be transferred sufficiently rapidly to maintain isothermal conditions. If the pressure at the downstream end of the pipe were Pw, the fluid there would then be moving with the velocity of a pressure wave, and therefore a wave could not be transmitted through the fluid in the opposite direction because its velocity relative to the pipe would be zero. If, at the downstream end, the pipe were connected to a reservoir in which the pressure was reduced below Pw, the flow conditions within the pipe would be unaffected and the... 

Compounds A and B are available in the off-gas stream from an absorption column at concentrations of 20 moles/m3 each. 14 m3/sec of this fluid is to be processed in a long isothermal tubular reactor. If the reactor may be assumed to approximate a plug flow reactor, what volume of pipe is required to obtain 80% conversion of species A ... 

The mass flow rate under adiabatic conditions is always somewhat greater than that under isothermal conditions, but the difference is normally <20%. In fact, for long piping systems (L/D > 1000), the difference is usually less than 5% (see, e.g., Holland, 1973). The flow of compressible (as well as incompressible) fluids through nozzles and orifices will be considered in the following chapter on flow-measuring devices. 

In principle, this is the same as for single-phase flow. For example in steady, fully developed, isothermal flow of an incompressible fluid in a straight pipe of constant cross section, friction has to be overcome as does the static head unless the pipe is horizontal, however there is no change of momentum and consequently the accelerative term is zero. In the case of compressible flow, the gas expands as it flows from high pressure to low pressure and, by continuity, it must accelerate. In Chapter 6 this was noted as an increase in the kinetic energy. 

In the years since 1940, a voluminous literature has appeared on the subject of two-phase cocurrent gas-liquid flow. Most of the work reported has been done in restricted ranges of gas or liquid flow rates, fluid properties, and pipe diameter, and has usually been specific to horizontal or vertical pipe lines. The studies have in most instances been isothermal when two components were being considered nonisothermal cases were almost entirely single-component two-phase situations. Reports of investigations of two-phase two-component cocurrent flow where one component is being transferred across the interphase boundary are nearly nonexistent. 

Air is flowing at the rate of 30 kg/m2s through a smooth pipe of 50 mm diameter and 300 m long. If the upstream pressure is 800 kN/m2, what will the downstream pressure be if the flow is isothermal at 273 K Take the viscosity of air as 0.015 mN s/m2 and assume that volume occupies 22.4 m3. What is the significance of the change in kinetic energy of the fluid ... 

Let us write the model of nonstationary flow distribution as applied to the problem of search for the maximum pressure rise at a given node of the hydraulic circuit at a fast cut off of the flow in one of its branches (or the largest drop at pipe break) provided that there is an isothermal motion of viscous incompressible fluid subjected to the action of the pressure, friction, and inertia forces (Gorban et al., 2006). find... 

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