Developments in Mathematical Simulation of Fluid Flow in Building Drainage Systems

Dr. D. P. Campbell, B.A., Ph.D.,

Drainage Research Group,

The School of the Built Environment

Heriot-Watt University

Edinburgh, EH14 4AS

, Tel +44 131 451 4618

Title:

Developments in Mathematical Simulation of Fluid Flow in Building Drainage Systems

Abstract

A mathematical model utilising the Method of Characteristics to solve the equations of continuity and momentum in a finite difference scheme forms a computer based simulation package (AIRNET). It can simulate conditions in building drainage, waste and vent (DWV) systems. Recent work extended AIRNET by including the effects of detergents and temperature. Other work examined the motive force of the water, extending it to include a proportion of the discharge appearing as droplets falling in the central air core as opposed to assuming it was due entirely to an annular film. This paper includes these additions and links to previous work on non dimensional analysis, to yield a more refined AIRNET model which includes the effects of detergents, temperature and droplet fraction. Significant refinement of the AIRNET model is described in this paper.

Practical Application

Accurate simulation of detergent dosed waste as described here extends the scope over which accurate simulations and code guideline comparisons can be made, to include ‘live’ domestic and industrial media from modern appliances and practices. This will have a significant impact on code generation and drainage system design practices, allowing optimisation for reliability, effectiveness and cost. This paper improves the current level of understanding of the hydraulic basis for the design requirements imposed on sanitation engineers and code bodies, and improved an existing computer simulation model in its capacity as a design tool.

KEYWORDS:

building drainage, remote sensing, fluid flow, stack discharge, modelling

UNITS AND ABREVIATIONS

Aarea

partial derivative

ddiameter

DCfree diameter of wet stack

DRGDrainage Research Group

DWVDrainage, Waste and Ventilation system (in buildings)

ffriction factor

FSforce due to shear stress

ratio of specific heats

wavelength of laser light

Lstack length

Qvolumetric flow rate

density

Rradius (of stack)

interface shear stress

uvelocity

Vtterminal velocity

Vmaxmaximum velocity

Background.

One of the most important roles of a drainage system is the isolation of the habitable space from the waste material and its associated effects, achieved by the provision of water trap seals at various points in drainage networks. Design guides aim to produce ventilated drainage networks which protect these trap seals from unwanted air pressures. These design guides are based on investigations conducted primarily at BRE from the 1950’s onwards, during which it was assumed that water enters a drainage system as unsteady flow and descends in a stack as an annulus inducing movement in the central air core, Pink [1]. This air movement manifests as a suction pressure in the network, which water traps must withstand. Current design guides and building codes protect trap seals from suction pressures over -375 Nm-2 Wise, [2] in the form of BS5572[3]. Formative work can be traced back to Wise [7], progressing through a series of site investigations involving multi-storey buildings by Lillywhite & Wise [4], in which the mechanism of drainage stack annular down flow was defined.

Recently, this area has become the subject of intensive investigation with a coherent series of SERC/EPSRC research grants. The first investigated the propagation of air pressure transients in building drainage vent systems. The equations describing the boundary conditions represented by common system components were identified by Swaffield & Campbell [6] and incorporated into an existing mathematical model (AIRNET). The AIRNET model is based on a finite difference scheme and utilises the method of characteristics as a solution technique to simulate drainage system operation via the equations that define unsteady partially filled full bore pipe flows and the boundary conditions represented by water traps and other common system components. In terms of model algorithm development, the relationship between the suction pressures developed in the stack and water downflow rates was found to be analogous to that of the operating characteristics of a fan by Swaffield & Campbell [7]permitting the model to simulate the interaction between the hydraulic driving force behind air movement and the modifying effects of the system components themselves. This was later made independent of laboratory based test systems by Swaffield, Campbell & Jack [8]. The role of water films formed by established downflows falling past side entries and acting as transient attenuators was also included, as was the effect of intermittent stack occlusion and subsequent air flow stoppage caused by branch flow as covered by Wise and Swaffield [9] and Swaffield & Galowin [10]. This investigation was limited by its application to a single pipe diameter of 100mm.Jack [11] extended the model by applying the established techniques to pipes in general, overcoming the 100mm-diameter restriction. Site investigations using multi-storey buildings and ‘live’ drainage systems were a feature of this investigation, as were greater wet and dry stack and horizontal branch lengths, multiple combinations of user-defined appliance discharges, and higher water flow rates.

Campbell and MacLeod [12,13] reviewed the assumption that airflow induction was due solely to cold, clean water falling as an annulus and acting as a driving force. A new model was developed which identified the changes associated with the two main detergent groups: anionic and cationic; the effect of temperature was also quantified. Campbell [14] developed the work further and found that a continuous fraction of the stack discharge persisted in the central air core and accounted for the contribution to airflow conditions made by it. Campbell [15] then applied the particle tracking velocimetry technique to validate the model. Campbell [16] continued the work with a dimensional analysis of the factors involved, which provided a potentially useful solution technique.

This paper extends the work and establishes a general solution for DWV networks, including boundary conditions, charged with detergent dosed water.

Stack Discharge Analysis

It has been assumed that the velocity profile of the annular flow in a DWV system stack is linear and that the airflow flow down the vertical stack is as a result of the no-slip condition between the film and the air. The estimation of the interface velocity of the annular film and the air core is of great importance as it describes the air-water interaction and driving force behind air entrainment and thus the pressures that result during stack discharge. It is normal practice to refer to the annular film attaining a terminal velocity. If a linear profile is assumed through the film then the interface velocity is given by:

However there has been no data to support this assumption.

If a falling film is considered having laminar flow the shear stress acting on the film interface is given by:

This is equal to the weight of the film that is a product of the specific weight, leading to:

The constants of the integration can be obtained by applying the boundary conditions to the liquid film.

Hence, and by differentiating, the expression for velocity distribution is given by:

The velocity gradient at the air/water interface is zero. Therefore the volumetric flow rate then becomes:

The average film velocity is given by dividing V by the film thickness to give:

The ratio of interface to mean velocity of the film is given by:

Therefore it follows that annular film interface is given by

This is an estimate for the laminar flow condition. In reality the flow is often turbulent in nature. This will modify the boundary layer profile which will be thicker and the mean velocity will be higher. This means that for turbulent flow:

This analysis makes an assumption that the velocity gradient at the interface of the water film and air is zero. However if this is not the case then the velocity gradient will be a function of the air flow rate:

So, the velocity gradient will change depending on the airflow however at this stage the above equation is assumed to be a good approximation.

Airflow analysis

A characteristic feature of annular flow is the interface between the air core and the film. It is not a smooth surface, but one with complex wave patterns[17]. As a result of this a single-phase air core does not exist. The work carried out in this paper supports these findings.

Previous work[12] outlined the classes of detergents covered in this investigation, i.e:

(i)anionic (common in soap powders, tablets and liquids)

(ii)cationic (used as fabric conditioners)

(iii)amphoteric (specialist blenders in de-greasers)

(iv)non-ionic (second most widely used in washing powders)

With further addition of detergents abrupt changes in the physical properties of the solution occurs at a specific concentration; surface tension, osmotic pressure and turbidity all change abruptly. The effect increases upto a point (critical micelle concentration (CMC)) and then remains constant. Since the CMC is reached at detergent concentrations far below those used domestically or industrially (a factor of 5 is common), the manufacturer recommended dosing levels were adopted throughout. All detergents are supplied with various builders, perfumes and carriers with the actual concentration of active ingredients commercially confidential and reported as lying within a range; i.e. 5-15%. In this case, the median value was used in calculating the volume required for the tests reported here. The actual detergent concentrations used (reported here as the weight to weight or w:w ratio) are as follows:

(i)anionic 0.023%

(ii)cationic0.030%

(iii)amphoteric0.018%

(iv)non-ionic 0.025%

Two temperatures were used in this work: ambient (18˚C) and warm (55˚C), as determined by measurement in a two-storey 1:1 scale model built in a laboratory, using standard uPVC fittings and standard fixtures, and mains cold water or 40l discharges of water dispensed at 1 l/s at 60˚C. This is the maximum temperature that could be experienced in practice from a operational industrial dishwasher and this was chosen to maximise the effect.

Friction factor and Core Flow Phases

Three flow zones are known to exist within the air core:

  • Liquid film with near 100% single phase water or water surfactant flow
  • Two phase liquid air mixture near film surface
  • Like zone 2 with more air and less water

The air movement is a result of the shear stress or friction between the total water movement, including water droplets, and the air. Within building drainage flow modeling it has been assumed that two single phases exist and that the air movement is due to the surface of the annular film.

Wallis & Hewit [18] concluded that for annular flow at low gas and liquid velocities the film surface is smooth and that there is little droplet entrainment and the friction factor at the film air core is approximately equal to the pipe roughness. However above a critical value of air and water movement the friction factor rapidly increases, the tops of the surface waves are sheared off to form droplets and the density of the core increases. Campbell & MacLeod [14] showed that this was a simplification, and that a more accurate force balance due to a falling annulus plus a significant fraction of the discharge in the form of droplets would then be:

The total shear stress is the sum of the shear stress at the film water interface and the shear stress of the droplets:

This gives the total shear stress for the two-phase flow that is the sum of the shear of the droplets and the annular film interface.

Therefore the total shear can be expressed after substitution as:

This relationship gives the pressure gradient over the wet stack length, and incorporates the contribution to the motive force applied by water droplets in the air core and integrates this with the distributed friction model.

Pressure Zones

It is recognised that there are pressure zones present in a drainage system. There will be suction pressure at the base of a dry stack, a localized pressure below a discharging branch above the curtain and a back pressure at the base of the wet stack. All are critical for determining the entire pressure profile of the network. Obtaining an accurate profile is necessary to ascertain trap seals response.

The suction pressure at the top of the wet stack zone 1 is the sum of the vent entry loss and the dry stack friction loss. This pressure will be dependent upon the venting design that is the vent piping length, piping material and diameter. It can be calculated from a loss factor for the entry loss and Darcy’s equation.

The pressure drop experienced at the branch Zone 2 as a result of the discharge can be calculated in the same way as the entry loss to the dry stack, except the loss factor K can be obtained from an equation developed by Jack [11]:

Where the K loss factor is given by the following equation:

And

It should be noted this equation was developed based on the fluid trajectory from the branch to the stack from scaled drawings. However when comparing the difference in the free area from the calculated fluid trajectory to that estimated from drawings, the difference was negligible and it was not necessary to modify the equation as it is a good approximation. As such, the above equations developed by Jack were used to calculate this pressure drop.

A region of positive air pressure occurs immediately above the base of the wet stack. This arises as the fluid detaches itself from the pipe bend and a curtain is formed. The entrained airflow has to push its way through the curtain and a positive pressure is experienced which then recovers to atmospheric pressure. This to is calculated from a loss factor K where:

Pseudo Friction factor

The pressure zones are as a result of air transients driven by the discharged fluid within the pipe work. The interaction with the fluid and the air can be described as a friction factor as developed by the Darcy’s equation which is expressed as:

Rearranged where the air velocity is relative to the film velocity gives:

Where is the mean air velocity of the air core and is the velocity of the annular water film at the air water interface. The friction factor equation also has an equivalent height term this height can be defined as: the length at which the annular film does work on the air core, or the length the air core does work on the annular film.

This length is thus different from the stack height as the stack may be connected to several floor levels in the building. Also the waste discharge doesn’t instantly do significant work on the air core. Only when annular flow is developed does significant work on the core occur. So an estimate of the equivalent height could be the distance from the stack base to the point at which annular flow is developed. The equivalent height is therefore dependent on the geometry of the junction, the flow rate and the pipe material.

The trajectory of the fluid from a junction could be calculated based on the radius of curvature of the branch, the discharge flow rate and the friction coefficient of the pipe material. Taking this analysis further, if the flow from the junction is looked at in more detail when the discharge fluid element hits the stack wall the fluid splits up and has a high degree of swirl. Some of the discharge rebounds off the pipe wall and hits the opposite wall at the same angle as the incident angle. From this approximation the height at which annular flow occurs can be estimated:

From this the point the fluid makes contact with the stack wall is given by:

Assuming the rebounded fluid from the wall goes to the other wall at the same incident angle then the height at which annular flow is developed can be approximated to be:

To obtain this value however the flow rate of the discharge from the junction has to be calculated using the Chezy coefficient for the pipe material. The equivalent height can now be estimated as:

As the water flow rate increases so does the equivalent height as annular flow is reached quicker and so there is greater height for the film to do work on the air. Enough data is now available to calculate the pseudo friction factor for the air transient propagation.

Operation of the Boundary Condition and AIRNET Model

A technique was developed by Swaffield & Campbell [6,7] which describes the movement of air in discharging DWV systems in terms of the equations of continuity and momentum:

r+u+=0