Pressure drop Darcy equation

For direct measurement from core samples, the samples are mounted in a holder and gas is flowed through the core. The pressure drop across the core and the flowrate are measured. Providing the gas viscosity (ji) and sample dimensions are known the permeability can be calculated using the Darcy equation shown below. 

The fundamental case for pressure filters may be made using equation 10 for dry cake production capacity Y (kg/m s) derived from Darcy s law when the filter medium resistance is neglected. Eor the same cycle time (same speed), if the pressure drop is increased by a factor of four, production capacity is doubled. In other words, filtration area can be halved for the same capacity but only if is constant. If increases with pressure drop, and depending how fast it increases, the increased pressure drop may not give much more capacity and may actually cause capacity reductions. 

Pressure drop in static mixers depends very strongly on geometric arrangement of the inserts. It is simply defined in relation to the pressure drop AP in an empty tube given by Darcy s equation ... 

For filter design and performance prediction it is necessary to predict the rate of filtration (velocity or volumetric flowrate) as a function of pressure drop, and the properties of the fluid and particulate bed. This can be achieved using the modified Darcy equation developed in Chapter 3. 

Estimation of the pressure-drop The system is designed to work within a given pressure limit thus, one needs a relation giving the pressure-drop in the column (per unit length). Darcy s law gives the relation of AP/L versus the mobile phase velocity u. However, the Kozeny-Carman equation is best adapted for laminar flows as described ... 

The single-phase friction loss (pressure drop) for these situations in chemical and petrochemical plants is still represented by the Darcy equation with specific limitations [3] ... 

If calculated pressure drop from inlet to outlet of line system (not including control or hand valves) is greater than approximately 10%, but less than about 40% of the inlet pre.s.sure Pi (pounds per. square inch gauge), the Darcy equation will yield reasonable accuracy when using a specific volume based on the average of upstream (inlet) and do vnstream (outlet)... 

The fnctio-nal resistance or pressure drop due to the flow of the fluid, hf, is expressed by the Darcy equation ... 

Because flashing steam-condensate lines represent two-phase flow, with the quantity of liquid phase depending on die system conditions, these can be designed following the previously described two-phase flow methods. An alternate by Ruskin [28] uses the concept but assumes a single homogeneous phase of fine liquid droplets dispersed in the flashed vapor. Pressure drop was calculated by the Darcy equation ... 

The most straightforward porous media model which can be used to describe the flow in the multichannel domain is the Darcy equation [117]. The Darcy equation represents a simple model used to relate the pressure drop and the flow velocity inside a porous medium. Applied to the geometry of Figure 2.26 it is written as... 

This equation defines the permeability (K) and is known as Darcy s law. The most common unit for the permeability is the darcy, which is defined as the flow rate in cm3/s that results when a pressure drop of 1 atm is applied to a porous medium that is 1 cm2 in cross-sectional area and 1 cm long, for a fluid with viscosity of 1 cP. It should be evident that the dimensions of the darcy are L2, and the conversion factors are (approximately) 10 x cm2/darcy C5 10-11 ft2/darcy. The flow properties of tight, crude oil bearing, rock formations are often described in permeability units of millidarcies. 

The main fundamental difference between open columns and packed columns is in the required pressure drop, for which Darcy s law provides a general equation ... 

The relationship between flow and pressure drop is expressed by Darcy s law [7,8], an empirical equation applied (starting in the last century) to numerous forms of porous media. This law may be thought of as the Ohm s law of fluid flow, another linear law of transport expressing, in this case, the flux density q of fluid (analogous to current density) in the form... 

A mathematical formulation of the transient behavior of pressure in an HDS reactor is, of course, extremely complex and. as yet, has not been analyzed. In the limiting case, when the catalyst bed is heavily plugged with solid deposits, the problem of pressure-drop calculations is very similar to the pressure drop through an oil reservoir.72 The well-known Darcy type of equation should be applicable in this case. 

Chemical engineers are familiar with the Fanning (or Darcy) friction factor,/, the Moody chart of/vs. Reynolds number, Rg, and how all of this fits together to calculate pressure drop for a given fluid flow in a given sized pipe. The friction factor is calculated from the Colebrook equation ... 

The basic pressure-drop equation (the Darcy fonuula) for flow thi ough a pipe can be wiitten as ... 

The friction factor is used in the Darcy-Weisbach equation for calculation of pressure drop for turbulent flow in an empty pipe. The mixer pressure drop is given by... 

Thonglimp s equation estimates the minimum particle fluidisation velocity and the particle entrained height and Ergun s law approximates the pressure drop across the filter. Darcy s law predicts the gas flowrate distribution through out the filter. 

In the above equation, L is the length over which the pressure drop p — p2 is to be calculated pi is the absolute pressure of the flow at an upstream point 1, and p2 is the absolute pressure of the flow at a downstream point 2 / is the Darcy-Weisbach friction factor that can be determined from the Moody Diagram y is the adiabatic exponent (equal to 1.4 for air) and Mi is the Mach number of the flow at the upstream point 1. In addition to the pressure equations, the following equation of state of ideal gas is also needed ... 

The third major factor in dynamics is packed bed flow behavior. The pressure drop in the adsorbent media is caused by drag due to fluid flowing through the interstices. For slow flow, the pressure drop, AP, across a bed of length Az and superflcial velocity v, can be represented by Darcy s equation ... 

Integration of the Darcy equation to obtain flow rate as a function of the pressure drop across the cake Ap, and the thickness L, or the volume of solids per unit area Overall material balances involving slurry, filtrate, and cake and determination of pressure and filtrate volume as a function of time. 

Equation (7,17) is called the Kozeny-Carman equation and is applicable for flow through beds at particle Reynolds numbers up to about 1.0. There is no sharp transition to turbulent flow at this Reynolds number, but the frequent changes in shape and direction of the channels in the bed lead to significant kinetic energy losses at higher Reynolds numbers. The constant 150 corresponds to = 2.1, which is a reasonable value for the tortuosity factor. For a given system, Eq. (7,17) indicates that the flow is proportional to the pressure drop and inversely proportional to the fluid viscosity. This statement is also known as Darcy s law, which is often used to describe flow of liquids through porous media. 

Perhaps the most important variable in the description of foam flow through porous rock is the mobility of the foam, the flow achieved for a given pressure drop. This quantity is defined as the simple ratio of the combined superficial flow rate to the imposed pressure gradient. This ratio is indicated in the first part of equation 2, in which the mobility, A, is given in terms of the combined flow rate, Q, the cross-sectional area of the sample, A, the pressure drop, AP, across the sample, and its length, L. As is well known, in the flow of an ordinary fluid through porous media, the mobility can be separated into two factors one has to do primarily with the properties of the rock, and the other, with the properties of the fluid. By using Darcy s relation, the mobility for the ordinary fluid is computed to be the ratio of effective rock permeability, k, to the fluid viscosity, n. 

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