Reduction of Environmental Impact of Drum Machine Washing

An improvement of efficiency of the horizontally rotating drum washing machine is possible by using a more open type of drum, essentially without suds in the annulus, by using a pump to wet the clothes during rotation and fall. Modelling and simulation are used to quantify these claims and further optimize the design of the horizontal washing machine. The flow of suds inside the deforming clothes at impact with the drum is calculated. The wash performance is shown to be largely proportional to the open perforation area in the drum. The traditional design uses 1/8 of drum area for the perforation holes. A significant reduction of water, detergent, electrical energy, and wash time, with parity in wash performance, provides a step towards a cleaner and more sustainable future.


Introduction
Wash performance is a balance between temperature, chemistry, time, and mechanical action [1] [2] [3]. Many of these factors have been researched, quantified, and used to optimize the wash performance to the satisfaction of the consumer.
Solid and oily dirt removal occurs when mechanical drag forces from the wash fluid exceed the interfacial retaining forces that attach the dirt to the fabric. For water on the outside, the water can continue to flow in the direction of the fall, almost perpendicularly to the surface of the drum, creating a faster flow, and hence, better and faster cleaning compared to an article that would hit a solid surface of the drum. The mesh of the drum gauze must be small enough to avoid trapping attachments on clothes (e.g. buttons, cords, strings.) In centrifuges, the drum is mounted often vertically rotating, in washing machines the drum may rotate horizontally or vertically. In the horizontal drum wash process, the rpm is so low (typically 25 -60 rpm) that the wet articles fall due to gravity and impact with the lower part of the drum to enhance the libera- In spinning and centrifuging, the rpm are so high that a large part of the liquid is expelled from the fabric through the holes by the centrifugal force on the To make the process more favourable in a horizontal axis washing machine, we propose to fill the drum holes and the annulus with air instead of water, by using less water and pumping the water more quickly away from the annulus.
That water would muffle the impact and thus, reduce the release of water when clothes impact the drum after the fall in a horizontal drum. Moreover, the drag on the drum is reduced, as the drum no longer rotates through the stagnant water in the annulus between the drum and the housing.
Less water means that for the same dose, the concentration of surfactant is increased, creating better wash-process efficiency, and more efficient batch rinsing. Alternatively, the dose may be reduced, still maintaining the same or even a higher chemical concentration, while fewer chemicals are consumed. Very high concentrations of surfactants and bleach lead to damage of dyes in clothes. For surfactants, a concentration of the order of the critical micellar concentration during the wash is optimum.
During the wash process, the water is re-injected, to re-wet the clothes before they fall again. The circulation speed (refresh rate) of the water is improved compared to conventional design, improving the speed of mixing of chemicals, faster heating, and improving the efficiency of the wash process. The water is preferably injected through the side surface of the drum, just beyond the point of fall contact, such that the clothes have sufficient time and can take up the water again on their way up to the next fall. By choosing the injection more or less in line with the direction of rotation, the beam helps to rotate the drum. The open design of the drum surface also allows the re-injection of the water with sufficient force (as a beam or spray) through the open drum surface to add to the cleaning efficiency, to add to the lift force, and to loosen the articles from the drum surface. The extra force of the injected water in the direction of the motion, as well as the absence of drag through water outside the drum, relaxes the design strength of the motor. The re-injected water (suds) acts against the centrifugal force of the rotation and the downward force of gravity. Injection of wa- ter through the open surface of the drum will expand the compacted layer and stretch the fabric before the next fall. Hence, articles will wet more efficiently than in the conventional design and the injected water will allow the articles to better relax their shape. The injection might be directed more in an upward di- The exact injection point/line will depend on the rotational direction of the drum. Maybe the 3 or 4 ribs usually seen in conventional designs for drum strengthening and to drag the clothes around can be removed. They reduce the wash efficiency: Clothes in contact with the ribs are rotated more towards the middle of the drum, and are more likely to fall due to reduced centrifugal force, thus releasing their water less efficiently.
As the annulus is mostly filled with air, if needed, small ribs can be mounted on the outside in order to strengthen the drum. Since the cleaning potential is greater for a larger inner drum radius and lower rpm, a maximum radius of the drum is recommended in the design of the drum.
The rougher surface of the mesh of the drum surface will create a more even drag force on the clothes as they fall onto the drum and are then dragged along with the rotation of the drum. The slip at the first moment of contact with the drum is reduced, reducing the surface wear of the articles and creating a deeper wash effect (slip only partially cleans the surface). The more even drag will distribute the wash effect more evenly over the whole contact time, more evenly cleaning the article, and more evenly distributing the washing force (the bend- The amount of water to effectively soak the articles is reduced from 7 -10 li-Journal of Applied Mathematics and Physics tres to say 3 -4 litres or less for a 5 kg wash. The chemical formulation detergent dose can be reduced by the same factor. Less water means proportional reduction of time and energy spent for heating. Also, foam inside the fabric may reduce water use.
The active circulation of the wash liquor will reduce stagnant areas and volumes where chemicals can accumulate without contributing to the wash process. Rinsing becomes more efficient, with shortened time and less water when the amount of water per rinsing step is reduced.

Modelling and Simulating Wash Performance in Drum Washing
The trajectory of a small article in a horizontal rotating drum is derived in Ap-

Single Phase Flow in Porous Media
The simulation of single-phase flow from gravitational compaction in deformable porous ("poroelastic") media is described in [4]. We will use a similar the-

A Simplified Wash Model
In the model, we assume for simplicity that the article hits a stationary drum perpendicularly in a one-dimensional motion in a vertical z-direction with a speed Δv. The article is stopped, while the water continues to flow for a while through the grating of the drum, but slowed down by the drag of the fabric. In a simple model, we will assume that the area in the maze of the drum is open and allows free flow of water. The z-coordinate is the height above the drum surface.
This single vertical z-coordinate is a simplification of the (x,y) trajectory near impact with the drum. At impact on the drum, an instantaneous pressure excess ( ) is created in the fluid at the surface of the fabric, driving the liquid through the drum. While the liquid drains, the fabric is contracted in the vertical direction and a strain is developed to support the pressure gradient S the strain δz/z, φ the porosity, and ε the compressibility of the fabric. The flow of liquid is directly changing the (liquid) porosity. The conservation of volume requires q z t Combining these equations, we find Hence, we arrive at a differential equa- Both permeability k and elasticity ε are functions of porosity φ . Simple generic functions in our porosity range are for permeability Carman-Kozeny and for elasticity the modified heuristic Van Wyk's equation . This is a nonlinear second-order partial differential equation (PDE) that can only be solved by numerical simulation for the proper initial and boundary conditions, but for small excursions in porosity which is a purely parabolic PDE. This leads to scaling Journal of Applied Mathematics and Physics Suppose we assume that the material drains from one side in a one-dimensional way and that the material is so thick, or times are so short, that the drainage disturbance does not reach the back of the fabric, within the time span of observation. Standard mathematics leads to a solution ( ) 0 This solution is familiar in diffusion and heat conductance modelling. Erf is the The amount of fluid flowing is Liquid flows in the direction of the lowest pressure.
To make liquid flow out of the porous medium through z = 0, the pressure on the outside must be lower than inside. Then The liquid continues to flow with time at a diminishing rate. This is the case because it was assumed that the disturbance has not (yet) reached the other side of the medium. In our case, the production of fluid at z = 0 leads to a compaction shortening of length For a set parameters that would represent cotton, e.g. 0 0.63 φ =

Simulation of Flow Improvement for an Open Gauze Type of Drum
We will use a simple explicit simulation with Gauss-Seidel iteration to show the impact of holes compared to impacts on open and closed gratings in a steady state.
We use a 3-dimensional simulation of creeping flow impact to simulate the effectiveness of holes and compare that with the effectiveness of a totally closed and completely open grating impact (which are essentially 1-dimensional simulations). If we assume that the parameter combination 2 .
i j k n i j k n i j k n i j k n i j k n i j k n i j k n i j k n s . For this step size, the porosity at a node after each time step is simply the average of the surrounding nodes.
When we simulate the behaviour after many time steps, a given porosity distribution at the boundaries, we approach a steady-state solution for porosity dis- x j =∆ * , y y k =∆ * , t n t = * ∆ ) with appropriate boundary conditions. The article is 2 × 2 × 2 mm 3 and falls in the z-direction on a grating. We will use the same wetted cotton parameters as before:   to 2 × 2 mm 2 increases the compression from 6.23% to 18.5% (steady-state, e.g. 12 to 13 ms) and from 3.46% to 11.77% in 0.63 ms wash time. An open area hole size increase with a factor 6 leads to a factor 3 in the speed of wash; this is significant, but not fully proportional to the area increase. If we add extra holes of the same size in between the existing holes, the wash speed will be nearly proportional to the number of holes.
In these simulations, the side walls did not allow vertical flow (porosity constant We observe compaction to 15.66%, significantly better than the 11.8% reached in Figure 6

Conclusions
The wash performance is more or less proportional to the perforated fraction of the drum area. A highly perforated drum may easily extend the open fraction to 1/2 of the area and will lead to a significant increase of the speed of the wash process. Pumping suds from the annulus and drum will further increase the wash performance and reduce the drag on the drum. Water, chemicals, and energy reduction in the wash process help to better preserve and sustain the environment. Reduced wash time is beneficial for the end-user.
Technical realization of these proposed improvements is left to dedicated engineers. Journal of Applied Mathematics and Physics

Appendices Appendix A. Theory: Article Movement in Horizontal Drum Washing
A mathematical model for the drum-type washing machine is the basis for our calculations; see also Figure 1 for a side view on the drum. The laundry is considered a point mass m describing a parabolic path. The drum is considered a horizontally mounted cylinder rotating on its axis with an angular speed ω (rad/s), frequency f = ω/2π, and a linear speed of a point on its drum surface of v c = ωR, where R the radius of the drum. A Cartesian coordinate system (x, y) is chosen with x horizontal, y vertical, and the origin in the axis to describe the coordinates of the trajectory of the point mass. Gravity acts vertically downward opposite to the direction of the axis y. The acceleration of free fall is g (9.81 m/s 2 ).
All chosen units conform to the MKSA system.
In the lower part of the drum, the point mass is pressed against the drum with a force The physically relevant root of this equation is The fall path has a total length of

Appendix B. Optimal Drum Design
The formulas found in Appendix A are used to optimize the drum design to maximize the velocity change of fabric at impact with the drum.
We are interested in the speed just before and just after impact. First, we will focus on the speed just before the impact. The point mass is then at the end of the parabolic trajectory and has at that moment a speed given by

( )
, sin . By rearrangement: . We may conclude that the g-factor = ω 2 R/g and a term Δv/(ωR) are the important dimensionless groups that determine the efficiency of a washing machine. The speed change as a function of drum radius and drum frequency for given acceleration of free fall of g = 9.81 m/s 2 is given in Figure B1.
For each frequency of the drum, there is a clear optimum in velocity change at a particular drum radius, and hence an optimal wash performance for horizontal axis drum-type washing machines at those particular conditions. We find the maximum velocity change at a given frequency by zeroing the first derivative of Figure B1. The impact velocity change (m/s) as function of speed of the drum (rpm) and the radius of the drum (m For each radius of the drum, there is a clear optimum in velocity change at a particular drum rotation speed, and hence an optimal wash performance for horizontal axis drum-type washing machines at those particular conditions.
The maximum velocity change at a given drum radius R is found by zeroing the first derivative of Δv to ω at a given and constant R The optimum trajectory is shown in Figure B2. We may compare these data in Table B1 with the data drum diameter/frequency for the main wash in commercial washing machines for in-home use. Figure B2. The optimum trajectory of laundry in a rotating drum washing machine of 48 cm internal diameter that should, for the optimum conditions at that radius, be operated at a rotation speed of 40.8 rpm. We observe that for the washing machines of which we have sufficient data, several are not operating near the optimal frequency for good wash efficiency for a small load.
We have found very simple formulas that help to optimize the design of horizontal axis washing machines to maximize the ultimate wash performance potential. The maximum velocity change corresponds to each cloth with the maximum impulse (Δp = mΔv) change. Since this impulse change is exerted in a very short time, this also corresponds with the maximum force F = Δp/Δt on the wet fabric, thus with the maximum relative speed of the wash liquor with respect to the fabric, and therefore with maximum potential wash performance.
Heavily loaded machines will create a bed of fabric, reducing the effective radius of rotation for the fabric at the top of the bed. The articles will then rotate and tumble backward, creating an effective lower rotation speed of the centre of mass of the articles compared to that of the drum, resulting in lower wash performance, and maybe another optimum rpm.

Appendix C. Multidimensional Simulation of Creeping Flow
Steady state 3-D porosity distribution for boundary conditions representing a cube of cotton with imposed porosity profile on boundaries.