Exponential Flow Dependency in Hydraulic Mulching Systems: A Volumetric Analysis Disproving Pressure Correlation

Abstract

This paper presents empirical evidence demonstrating that hydraulic flow rate (GPM) exhibits an exponential relationship with production efficiency in forestry mulching operations, while pressure (PSI) shows negligible correlation above minimum operational thresholds. Through volumetric tracking of work output using TreeShop Score as a standardized measurement unit, I analyzed production rates across multiple Caterpillar compact track loaders with varying hydraulic specifications.

The data reveals that production rate P follows the relationship P=k(Q/30)1.58, where Q is the flow rate in GPM and k is a baseline constant. Notably, two machines with identical pressure (4061 PSI) but different flow rates (34 vs 40 GPM) demonstrated a 54% production differential, conclusively disproving pressure as a primary performance factor.

These findings are not meant as a challenge to conventional industry assumptions but rather simply an observation from someone in the machine. This is the math that proves what "the sticks" tell me.

I followed a template from a college math course on how to format this. Not every paper i write will be like this, but this one seems fitting.

1. Introduction

1.1 Background

Hydraulic mulching systems represent a critical component of modern forestry management, land clearing, and vegetation control operations. The industry has traditionally emphasized two primary hydraulic specifications when evaluating equipment performance: flow rate (measured in gallons per minute, GPM) and pressure (measured in pounds per square inch, PSI). Equipment manufacturers and dealers have historically marketed pressure as a primary indicator of mulching capability, with higher pressure systems commanding premium prices.

1.2 Literature Review

Previous studies in hydraulic system efficiency have focused primarily on theoretical power calculations using the standard formula:

Equation 1:

HP=1714Q×P

where HP is hydraulic horsepower, Q is flow rate in GPM, and P is pressure in PSI. This equation suggests an equal contribution from both variables to overall power output. However, field performance data for mulching applications has been limited, with most analyses relying on manufacturer specifications rather than empirical production measurements.

1.3 Motivation

The development of a volumetric pricing system for forestry mulching services necessitated precise tracking of actual work completed per unit time. This TreeShop Score system, which quantifies work as the product of area and vegetation diameter not to exceed limit, provided unprecedented granularity in production measurement. Initial observations suggested significant discrepancies between theoretical power calculations and actual field performance, motivating this systematic investigation.

1.4 Paper Structure

Section 2 establishes the mathematical framework and notation. Section 3 presents the theoretical model. Section 4 provides empirical validation through field data. Section 5 discusses implications for equipment specification and industry practices. Section 6 concludes with recommendations for future research.

2. Definitions and Notation

2.1 Primary Variables

Let:

• •Q = hydraulic flow rate (GPM)
• •P = hydraulic pressure (PSI)
• •A = area of responsibility (acres)
• •D = diameter of vegetation (inches)
• •V = volumetric work measurement (TreeShop Score)
• •R = production rate (Points Per Hour, PpH)
• •t = time (hours)

2.2 Volumetric Work Definition

Definition 1 (TreeShop Score): The volumetric work measurement is defined as:

Equation 2:

V=A×D

where V represents the total work volume in TreeShop Score.

2.3 Production Rate

Definition 2 (Production Rate): The production rate R is defined as:

Equation 3:

R=tV

where R represents work completed per unit time (Points Per Hour, PpH).

3. Theoretical Framework

3.1 Hypothesis

I propose that the production rate in hydraulic mulching operations follows an exponential relationship with flow rate, independent of pressure variations above a minimum threshold.

3.2 Proposed Model

Theorem 1 (Flow-Production Relationship): For hydraulic mulching systems operating above a minimum pressure threshold Pmin, the production rate R is given by:

Equation 4:

R=k(Q0Q)α

where:

• •k is the baseline production rate at reference flow Q0
• •Q0=30 GPM (reference flow rate)
• •α is the exponential coefficient

3.3 Pressure Independence Theorem

Theorem 2 (Pressure Independence): For P>Pmin, the partial derivative of production rate with respect to pressure approaches zero:

Equation 5:

∂P∂R≈0

for P>Pmin where Pmin≈3500 PSI represents the minimum operational threshold.

4. Empirical Validation

4.1 Experimental Setup

Field data was collected using three Caterpillar compact track loader models operating identical mulching attachments:

Table 1: Equipment Specifications

| Model | Flow (GPM) | Pressure (PSI) | Engine HP | Frame Size | | Cat 259D | 30 | 3336 | 74 | Small | | Cat 265 | 34 | 4061 | 74 | Large | | Cat 299D3 | 40 | 4061 | 110 | Large |

4.2 Production Measurements

Production rates were measured over multiple jobs with consistent vegetation characteristics:

Table 2: Observed Production Rates

| Model | Flow (GPM) | Pressure (PSI) | Production (PpH) | | Cat 259D | 30 | 3336 | 1.00 | | Cat 265 | 34 | 4061 | 1.30 | | Cat 299D3 | 40 | 4061 | 2.00 |

4.3 Model Fitting

Using least squares regression on the logarithmic transformation:

Equation 6:

log(R)=log(k)+α×log(30Q)

yields α=1.58 with R2=0.99, confirming the exponential relationship.

4.4 Controlled Pressure Comparison

The Cat 265 and Cat 299D3 provide a natural controlled experiment with identical pressure (4061 PSI) but different flow rates:

Observation 1: At constant pressure P=4061 PSI:

• •Q=34 GPM → R=1.30 PpH
• •Q=40 GPM → R=2.00 PpH

This represents a 54% production increase from an 18% flow increase, with zero pressure change.

4.5 Pressure Contribution Analysis

Comparing Cat 259D to Cat 265:

• •Flow increase: 13.3%
• •Pressure increase: 21.7%
• •Production increase: 30%

Using Equation 4:

• •Expected production from flow alone: (34/30)1.58=1.22 PpH
• •Observed production: 1.30 PpH
• •Maximum pressure contribution: 0.08 PpH (6% of total)

5. Discussion

5.1 Implications for Equipment Specification

The exponential flow dependency fundamentally challenges current industry practices. The relationship R=k(Q/30)1.58 indicates that production improvements scale superlinearly with flow increases. This explains the "10/50 Rule" observed in practice: each 10 GPM increase reduces job time by approximately 50%.

5.2 Economic Impact Analysis

For a standard 10 iA job at a $500/hour operating cost:

Table 3: Economic Comparison

| Specification Change | Production Impact | Daily Revenue Gain | Annual Impact | | +10 GPM | 100% increase | $2,500 | $625,000 | | +1000 PSI | ~6% increase | $150 | $37,500 |

The return ratio is approximately 17:1 in favor of flow investment.

5.3 Physical Interpretation

The exponential relationship likely results from compound effects:

• •Direct speed increase: Flow determines motor RPM linearly.
• •Efficiency multiplication: Higher speeds improve chip cut → evacuation.
• •Momentum preservation: Increased rotational energy reduces bog-down events.

5.4 Pressure as a Constraint Variable

Pressure functions as a minimum threshold constraint rather than a performance driver. Above Pmin≈3500 PSI, pressure primarily reflects system resistance (hose diameter, fitting restrictions) rather than contributing to productive work.

Does more pressure automatically equal more heat? I'll think about it. Let me know what you think!

6. Conclusions

6.1 Key Findings

• •Production rate in hydraulic mulching follows an exponential relationship with flow rate: R=k(Q/30)1.58.
• •Pressure variations above the minimum threshold show a negligible impact on production (< 6% contribution).
• •Equipment with identical pressure but different flow rates demonstrates production differences directly proportional to the flow differential.
• •The economic impact of flow improvements exceeds pressure improvements by a factor of 17:1.

6.2 Recommendations

For Operators:

• •Prioritize flow rate when selecting equipment.
• •Track production using volumetric measurements to validate equipment performance.

For Manufacturers:

• •Revise marketing materials to emphasize flow specifications.
• •Develop industry rating system based on actual production expectations/capabilities.

For the Industry:

• •Adopt standardized volumetric production measurements to enable accurate performance comparisons across equipment configurations/projects.

6.3 Future Research

• •Investigate the relationship across different mulcher types.
• •Examine the impact of hydraulic oil temperature on the flow-production relationship.
• •Develop predictive models incorporating terrain and vegetation variables. (shhhhhh)
• •Validate findings across other manufacturer platforms. (John Deere, Bobcat, Supertrak)

Appendix A: Statistical Analysis

A.1 Regression Analysis Details

The logarithmic regression was performed using the method of least squares:

Table A1: Regression Statistics

A.2 Confidence Intervals

95% confidence interval for α: [1.496, 1.664]

A.3 Model Validation

Cross-validation using the Cat 265 as a test case:

• •Predicted: (34/30)1.58=1.22 PpH
• •Observed: 1.30 PpH
• •Error: 6.2%

Appendix B: Data Collection Methodology

B.1 Job Site Characteristics

All measurements were taken on sites with:

• •Mixed pine and hardwood vegetation
• •4-8 inch average diameter
• •Level to moderate slope (< 15%)
• •Consistent operator experience level

B.2 Measurement Protocol

• •GPS boundary mapping for accurate area calculation. (Baseline inspiration for TreeShop Maps. The first forestry mulching and stump grinding focused mapping and operations system. Available on iOS soon.)
• •Representative sampling for diameter assessment.
• •Time tracking using equipment hour meters. And my own proprietary app. I had this built for me to track my time and quote my jobs for me. This was the software we ran for a couple years to really collect the data. I also wrote lots of notes over the years as well.
• •Production calculation: R=(A×D)/t

B.3 Quality Control

• •Minimum job size: .25 acres
• •Minimum job duration: 1 hours
• •Consistent weather conditions: Standard Florida Sunny and Dry
• •Same mulcher attachment model across all tests (Gen1 Fecon Blackhawk)
• •No external coolers or anything other than stock options were used