# Simulation Tests of a Cow Milking Machine—Analysis of Design Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}Simulink. The authors modelled vacuum in the claw’s milk chamber for 0, 1, 2, and 3 mm changes in air flux diameters [54,55] and developed a simulation model for a system controlling milk flow in the collector column of an autonomous milking cluster [56]. A control system that will adjust the area of the aerating hole in such a way as to maintain a minimum vacuum drop throughout the milking procedure when the milk flow rate is changing was proposed [57].

## 2. Materials and Methods

#### 2.1. Simplified Structure of the Vacuum Drop Model

_{M}is the sum of the reduced velocities of milk and air.

- Linear velocity is the sum of the velocity of the milk/air mixture with coefficient characterising a turbulent flow (1.2 u
_{M}) and rise velocity of a single bubble_{∞}in a quiescent liquid; - The reduced air velocity requires the actual conditions to be lowered to normal conditions (a change in pressure causes a change in gas volume);
- The rise velocity of a single bubble in a quiescent liquid is the function of Archimedes (Ar) and Eötvös (Eo) numbers, which, due to the air density, depend on pressure p in the long milk tube;
- Volumetric air coefficient α
_{pdpm}in the long milk tube is the quotient of the reduced velocity and linear velocity (calculated relative to the tube walls); - Pressure in the milking claw is equal to operating pressure p
_{o}plus drop Δp_{kol}.

_{o}+ Δp

_{kol})/2, and taking the assumed relationships between the reduced velocity of the (milk/air) mixture and volumetric air coefficient α

_{pdpm}into account, Majkowska [63] determined a simplified Bernoulli’s equation in the following form:

_{∞}is calculated using the following formula:

_{m}, Q

_{p}—flow of milk [kg·min

^{–1}] and air [m

^{3}·h

^{–1}];

_{dpm}—length of the milk tube [m];

_{o}—operating pressure [kPa];

_{N}—atmospheric pressure in normal conditions, 100 [kPa];

_{kol}—pressure in the claw’s milk chamber, [kPa];

_{pdpm}—volumetric air coefficient in the long milk tube [-];

_{m}, ρ

_{p}—density of milk and air [kg·m

^{−3}];

^{−1}];

_{m}—flux in milk mass (milk-like liquid) flowing in the long milk tube, [kg min

^{−1}];

^{−1}];

_{∞}—velocity of rising a single bubble in a still liquid, [m s

^{−1}];

_{dpm}—cross-sectional area of long milk tube, [m

^{2}].

- When analysing the value of derivative dv
_{∞}/dp for g = 9.81 ms^{−2}, D = 0.016 m, ρ_{m}= 1030 kg m^{−3}, T = 288 K, it was found that, for pressure variations ranging from 48,000 to 52,000 Pa, the derivative is approx. 0.5315 × 10^{−9}and is almost constant. A variation in diameter D does not change this relationship. - The insensitivity of the rise velocity of a single bubble to pressure variations in the analysed task results from the values of milk and air density. Milk density is ρ
_{m}= 1030 kg m^{−3}at 288 ÷ 293 K, whereas air density at 288 K and p = 101.3 kPa is ρ_{p}= 1.225 kg m^{−3}[64]. The ratio of these values is approx. 1000:1. As a result, in terms of the vacuum drops observed in the long milk tube and, thus, in terms of variations in operating pressure p_{o}, air velocity v_{∞}is almost constant.

_{kol}= x and v

_{∞}= c, and taking into account that p

_{N}= 100 kPa = 10

^{5}Pa, expression (2) takes the following form:

_{∞}= c, polynomial is obtained:

_{w}of polynomial (10) has the form:

#### 2.2. Verification of a Model with a Simplified Structure of Vacuum Drops in the Claw

#### 2.3. Sensitivity of a Model with a Simplified Structure of Vacuum Drops in the Claw

_{s}to A

_{s}+ΔA

_{s}, the status of the model will change from x to (x + Δx + o(║A

_{s}║) ). The drop model equation meets the assumptions of the implicit function theorem. A set of values of polynomial W(x) was analysed for coefficients that were the functions of preset parameters A

_{s}and input variables U

_{s}present in a milking cluster (vectors A

_{s}and U

_{s}). Elements of vector U

_{s}belong to set Ω

_{s}:

_{s}belonging to domain Ω

_{s}(12), the polynomial includes real roots. At each point, the polynomial is a function of class C

^{1}. Given the simple form of function (10), the following derivatives were calculated analytically:

_{0}, A

_{s}

_{0}, U

_{s}

_{0}], the derivative of the function defining an explicit form of variable x relative to the parameters composing vector A

_{s}is expressed by the following formula:

## 3. Results and Discussion

_{s}

_{0}, U

_{s}

_{0}, Q

_{m}) referred to the values of drops x(A

_{s}

_{0}, U

_{s}

_{0}, Q

_{m}), and are expressed as a percentage. The sign of the differential indicates the direction of changes, while its absolute value shows the strength of the impact of the change in parameter A

_{si}on changes in value x at point (A

_{s}

_{0}, U

_{s}

_{0}, Q

_{m}). Quotient $\frac{\Delta x}{x}/\frac{\Delta {A}_{si}}{{A}_{si}}$ is often used as a measure of model sensitivity. In this paper, ten percent changes in the parameters were analysed. The analysis of the results presented later in the paper will concern the expression $\frac{\Delta x}{x}*100\%.$ The sensitivity of model (9) was tested for points from the allowable set Ω

_{s}(12).

_{0i}, given in the legend below. A point in the legend is clearly identified with a colour and an index. The value of the differential (Figure 4) is calculated at the point with index “i”.

_{0i}= (Q

_{pi}, D

_{i}, H

_{i}, l

_{dpmi}, p

_{oi}, ζ, λ where q

_{m}= 10 kg min

^{−1}, l

_{dpm}= H

_{i}+ 0.3 m

*****${P}_{0i}$ = [${Q}_{\mathrm{pi}}$; 0.018; 1.9; 2.2; 49.3; 0.5; 0.03] Q_{pi} = 0.2 + 0.1(i − 1) [m^{3} h^{−1}], i = 1, …, 9 |

*****${P}_{0i}$ = [0.4; ${D}_{i}$; 1.9; 2.2; 49.3; 0.5; 0.03] D_{i} = 0.014 + 0.001(i − 1) [m], i = 1, …, 9 |

***** ${P}_{0i}$ = [0.4; 0.018; ${H}_{i}$; ${l}_{\mathrm{dpmi}}$; 49.3; 0.5; 0.03] H_{i} = 0.2 + 0.2(i − 1) [m], i = 1, …, 10 |

*****${P}_{0i}$ = [0.4; 0.018; 1.9; 2.2; ${p}_{\mathrm{ri}}$; 0.5; 0.03] p_{oi} = 49 + 1(i − 1) [kPa], i = 1, …, 10 |

_{m}is a value set for the actual milking phase. In practice, this value is not constant. Similar calculations were performed for different milk mass fluxes. Change q

_{m}does not alter the opinions on the model’s sensitivity to changes in the parameters. The observations point to the following relationships between the design parameters of the milking machine and the milking process:

- At lower heads of the milk/air mixture, the model is more sensitive to variations in parameters ζ and λ (Figure 5, black colour).Local mechanical energy losses in the pipes are caused by various obstacles located in the tubes. The values of local loss coefficients ζ should be determined empirically based on measurements; these values are affected by, for example, pipe curvatures, kinks in tubes, an abrupt increase or decrease in the pipe cross-section, etc.Linear loss coefficient λ depends on two parameters: Reynolds number (Re) and relative roughness of a pipe (e), which is a non-dimensional parameter. Relative roughness is defined as: e = k/d, where: k—roughness (equivalent sand-grain roughness); [m], d—inside diameter of a pipe [m]. For isothermal turbulent flows with a practical importance for water supply pipelines (for Re > 4000), the flow can exist in three zones: A—hydraulically smooth piping zone, λ = f(Re), B—transition zone (with variable roughness), λ = f(Re, e), C—quadratic drag zone, λ = f(e);
- An increase in diameter D reduces the sensitivity of the system to a change in parameters (Figure 5, green colour). For the long milk tube diameter of 0.014 m, the sensitivity of the model is approx. 6%;
- A decrease in air flow reduces sensitivity (Figure 5, red colour); the course of vacuum drops as a function of inlet air flux (Figure 5, red colour), which indicates the existence of flux Ω*pi at which the drop is minimal (approx. 15 kPa in the diagram), and the sensitivity of the model around that point is approx. 3%;
- Operating pressure does not alter the sensitivity of the model to changes in parameters ς and λ (Figure 5, blue colour).The analysis presented above applies to point P
_{0i}= [0.4; 0.018; 1.9; 2.2; 49.3; 0.5; 0.03].In the entire allowable domain Ω, the nature of the relationship is similar.

## 4. Summary and Conclusions

- A change in local loss coefficient ς and linear drag coefficient λ in the long milk tube will have a lower impact on vacuum drops if a smaller flux of inlet air Q
_{p}, a higher head of the air/liquid mix H, and a higher diameter of the long milk tube D are used; - Operating pressure does not affect these changes;
- The model is insensitive to changes in parameter T: ten percent changes in temperature T caused changes in the drops to be lower than one percent.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Relative errors of vacuum pressure drops calculated by model (1) and simplified model (10) for D = 0.016 m, H = 0.4 m, l

_{dpm}= 0.7 m, p

_{o}= 58.0 kPa and ξ = 0.5, λ = 0.03 for selected values of the milk stream in the range from 2 to 10 kg min

^{−1}.

**Figure 2.**Relative errors of vacuum pressure drops calculated by model (1) and simplified model (10) for D = 0.016 m, H = 1.9 m, l

_{dpm}= 2.2 m, p

_{o}= 49.3 kPa and ξ = 0.5, λ = 0.03 for selected values of the milk stream in the range from 2 to 5 kg min

^{−1}.

**Figure 5.**Relative value of the differential, calculated at points ([P

_{0i}, ΔP

_{0i})), where: ΔP

_{0i}= (0, 0, 0, 0, 0, 0.003, 0.05).

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Golisz, E.; Kupczyk, A.; Majkowska, M.; Trajer, J. Simulation Tests of a Cow Milking Machine—Analysis of Design Parameters. *Processes* **2021**, *9*, 1358.
https://doi.org/10.3390/pr9081358

**AMA Style**

Golisz E, Kupczyk A, Majkowska M, Trajer J. Simulation Tests of a Cow Milking Machine—Analysis of Design Parameters. *Processes*. 2021; 9(8):1358.
https://doi.org/10.3390/pr9081358

**Chicago/Turabian Style**

Golisz, Ewa, Adam Kupczyk, Maria Majkowska, and Jędrzej Trajer. 2021. "Simulation Tests of a Cow Milking Machine—Analysis of Design Parameters" *Processes* 9, no. 8: 1358.
https://doi.org/10.3390/pr9081358