The process in question uses a centrifugal fan pump to convey a 1000 gallon per minute flow containing 2% finely ground wood pulp to a head box...which in turn is supposed to ocnvert turbulent flow to laminar flow and then deposit the water & pulp slurry on a high speed moving wire in order to form a wet fiber mat.
However, the head box that we have unfortunately does not create laminar flow --- it instead creates flow vectors that direct a high percentage of the slurry flow from the center of the moving wire to the edges of the moving wire leaving us with a dimensionally imperfect fiber mat that deposits more fiber on the edges of the moving wire form than in the center ---- this causes numerous downstream quality and physical sheet strength properties.
Rather than replace or completely re-design the head box, which would be extremely expensive and time consuming, we crafted what can only be called "flow tubes" that when welded together as a unit and placed directly in front of the turbulent flow vectors from the head box, seem to actually straighten out the flow patter and ultimately produces laminar flow.....which is what we want!!!
Please see the attached diagrams that I have put together which show the "plan view" of the existing head box and flow pattern and then the plan view and cross-sections of the flow tube device that I mentioned above.
Here is the question....is there a way in which the flow patterns can be empirically modeled or derived mathematically?
In otherwords, can math and/or science (e.g. fluid dynamics) prove that this "flow tube" device can actually work?
Second, is there a flow equation that can be used to calculate the theoretical flow in each flow tube given the 1000 gallon per minute flow at a 2% consistency???
Any thoughts, direction, or help in deriving a more empirical understanding of how this device should work...or otherwise straightening out the turbulent flow in my head box would be very greatly appreciated.
Please see attached file.
I have done a lot of readings on this topic mainly from the technical Handbook of paper making science and technology. The "issue" you have is hypothetical. In fact, headboxes with nozzles are standard configuration these days in paper mills.
I will now explain your questions and provide the reference in the attached file.
1. Is there a way in which the flow patterns can be empirically modeled or derived mathematically??? In other words, can math and/or science (e.g. fluid dynamics) prove that this "flow tube" device can actually work?
Yes, engineering science has proved that it is necessary to design the network of multiple flow tube, or nozzles if you like in the headbox so that the flow pattern is uniform across the cross-direction (CD).
The many factors that affect the uniformity of fibre mat distribution on the web include:
a. Lip opening: The lip opening determines the initial thickness of the jet. The ideal jet is perfectly flat, which requires that it contains no turbulence.
b. Speed and flow direction: the mix jet flow requirements are that the speed and flow direction are constant across the whole width of the machine and that also the jet has a constant thickness in the CD.
c. Pressure distribution in headbox: It is particularly important to maintain uniform pressure profile across the CD so that uniform fibre web can be formed.
d. Design of nozzle, including diameter of nozzle, contract ratio (inlet diameter/outlet diameter). This is important to maintain uniform flow and control the flowrate in the nozzle.
e. Consistency of fibre suspension: Normally in the mill I'm involved with, the consistency ranges between 0.5% to 0.6%. This allow better mixing degree in the headbox. You may consider lower your value.
2. Is there a flow equation that can be used to calculate the theoretical flow in each flow tube given the 1000 gallon per minute flow at a 2% consistency?
Yes. However, different headbox design will require different model of calculations. The model will take into account all the above factors and it require a fulltime process engineer to establish the model and validate it with mill data. In the absence of headbox configuration and design details, that is all I can tell you.
Nevertheless, I can provide the reference attached (from the handbook) which could perhaps help you more with understanding this issue.
Hope that this gives you more insight into the issue. Do not hesitate to come back if you have further questions.
Below is a copy of the Word file.
The main function of the headbox is to distribute the mix evenly across the width of the wire section. This means, for example, that the flow from a pipe with a diameter of 800 mm shall be transformed into a 10 mm thick and 10 000 mm wide jet, with absolutely the same flow rate and flow direction at all points across the width, as indicated in Fig. 10.
Figure 10. Feed pipe for mix and cross-section of jet from headbox (not to scale).
Table 4. Typical jet thickness [mm] in industrial forming.
A simple equation that relates the required lip opening h [mm] to form a product of basis weight w [g/m2] from a mix concentration of c [g/L] is given below:
where R is the retention factor. This equation presumes that jet speed equals wire speed.
The flow transformation by the headbox from the incoming pipe flow to the delivered plane jet takes place in mainly three steps:
- The cross-direction distributor makes a first distribution of the mix across the machine width.
- Pressure drop elements are introduced to even out the CD flow profile.
- A headbox nozzle generates the final jet.
6.2.1 Cross-direction distribution
A modern cross-direction distributor usually consists of a tapered header, a channel that runs across the whole headbox width, from which discharge takes place successively through holes in the channel wall25 (see Fig. 11).
Figure 11. CD distributor with main flow Q, discharges D Q, and overflow q.
If a perfect headbox jet is to be delivered to the wire section, the mix flow DQ through each discharge hole must be equal. This means that the static pressure along the CD distribution channel must be kept constant. The friction pressure drop along the channel must therefore be compensated for by a corresponding pressure rise.
This pressure rise can be achieved by transforming part of the velocity energy in the flow along the channel into static pressure by successively reducing the flow velocity. The channel then functions in principle like a diffuser, even though the cross-sectional area A(x) decreases in the flow direction, in contrast to the case in a conventional diffuser. This is possible since the volumetric flow along the channel gradually decreases because of the discharge flows DQ. The following equation describes the flow velocity u(x) along the CD channel.
where n(x) is the number of flows DQ discharged before position x. The cross-sectional area A(x) along the channel changes so that, for given values of input flow Q and overflow q, a constant static pressure is obtained along the whole channel. It will not be possible, however, to maintain this constant pressure with other flow rates Q. If there is a pressure difference between the inlet and outlet, the overflow rate q can be adjusted so that pressure agreement is attained. Some pressure deviations inside the channel can nevertheless still remain.
The distance between the individual discharge holes should be so large that bridging of fibers across two adjacent holes is avoided. Such fiber piling would lead to the build-up of detrimental fiber flocs.
The discharge holes feed a tube bank or drilled plate. The higher the pressure drop is along the tubes, the smaller the differences will be between the individual discharge flows DQ caused by variations in static pressure along the CD distribution channel. To achieve a high pressure drop, the local flow velocity must be high, which means that the flow area in the tube bank should be relatively small. This means that the (relative) open area at the inlet of the tube bank, i.e., the ratio of the total area of the discharge holes to the channel wall area is small (in the order of 10 %). To feed a downstream chamber from only 10 percent open area would however generate unacceptable flow instabilities. To improve the stability of the downstream flow, the relative open area at the outlet from the tube bank therefore may be increased considerably.
This can be done in two alternative ways:
- The tubes or holes are expanded with gradual or sudden increases in diameter, so that the flow cross section is successively increased.
- The tube centers are brought together, so that the solid area ...
Head box issues in a pulp mill are discussed. The numerous downstream quality and physical sheet strength properties are determined.