Mathematical Model Of Process Research Articles

CERCETĂRI PRELIMINARE PRIVIND REALIZAREA UNUI MATERIAL COMPOZIT, BAZAT PE MATRICE DE NĂMOL ȘI DEȘEURI AGRICOLE, CA MATERIALE DE UMPLUTURĂ

The article presents the results of preliminary experiments necessary for the foundation of an experimental method that ensures the design and manufacturing technology for a category of composite materials with a mud matrix and agricultural waste insertion. The resources and physico-chemical properties of the mud matrix as well as the resources and physical properties of potential agricultural waste constituents of the insertion in the composite material are presented. The technological variant for the manufacture of the finished product is also presented, consisting of a compression system of the composite mixture in a paste state until solidification with moisture removal (dehydration). From a large number of parameters characterizing the experimental system, few have been varied (insertion concentration in the material, height of the raw material column in the press, raw material density, pressing force), while the quality parameters of the process have only been considered for the capacity of dehydration and material densification. The statistical analysis of the results shows that for the capacity of dehydration and densification, the main input parameters that influence are the initial moisture and initial volume of raw material. Results are obtained that allow the development of a mathematical model for the technological process of manufacturing the composite material. The purpose of the research described in this article is to determine the feasibility and feasibility conditions of the material in its possible variations. This means identifying the insertion concentration and moisture intervals of the material before extrusion, possible. The upper and lower limits of these parameters are sought, so that the material exiting the die does not immediately, or over time, disintegrate, and at the same time, the extrusion process is possible with a reasonable energy consumption.

Mathematical modeling of the technological process of twisting a waveguide

Twisted waveguides are part of waveguide paths of antenna- feeder devices of spacecraft. They have high requirements for accuracy and reliability. They are made of profiled rectangular pipes. The twisting of waveguide pipes with a cross- sectional size of less than 11x5.5 mm is carried out in the device using a rigid calibration mandrel. The calculation method determined the main parameters of the technological process of twisting the waveguide tube (torque M, drawing force Pпр). When twisting workpieces with cross sections according to GOST, the torques (M) and drawing forces (Pпр) from the cross- section sizes vary according to a nonlinear relationship. Twisting the workpiece from M4 requires less torque (M) and drawing force (Pпр) than for the workpiece from M1. By mathematical modeling, it is possible to select the optimal technological mode in advance, thereby ensuring high quality of waveguide twisting manufacturing. To substantiate the calculation results, the methods of control of technological parameters and quality control of the product manufacturing were considered. A problem has been found in measuring the torque and pulling force when twisting the workpiece. Therefore, the pulling force should be measured from the machine. A method for measuring torque using a non- contact torque sensor is proposed. The manufacturing quality is supposed to be carried out by the STRESSVISION mechanical stress scanner, and electrical tests.

Bir Direnç ve İki Kapasitörlü Lineer Bir RC Devresi için Cauchy Probleminin Fourier Serisi Cinsinden Çözümü

As it is known, solutions obtained in terms of Fourier series are widely used for more detailed analysis of mathematical models defined for different engineering problems. In this study, for the mathematical analysis of transient events in active RC based linear electronic circuits, the Cauchy problem is defined and solved in terms of Fourier series. For this reason, first of all, the mathematical model needed for the mathematical definition of the investigated problem was created. Based on the defined mathematical model, a quadratic differential equation for the linear RC electric circuit is obtained and the voltage and current changes are analysed for simple special cases using the Fourier series. As a result, mathematical modelling of transient processes occurring in linear RC circuits and mathematical problems defined as the Cauchy problem for such circuits have been solved analytically. It is thought that the results of the research will contribute theoretically and practically to the solution of the problems that arise in the study and design automation of different electronic circuits containing nonlinear circuit elements.

Исследование аэрогазодинамических процессов переноса радона из горных массивов

Introduction. Drying tailings are a source of gaseous chemicals, in particular radon222, which is carried by air currents along the slopes of the gorge. These processes need constant monitoring. Purpose of research. Establishment of climatic conditions most influencing the measurements of mountain radon and its dispersion in the atmosphere. Research methods and materials. The data of weather stations located in the Alagir mountain gorge and the data of the mathematical model of this gorge were used to establish the characteristic wind roses at the points of the gorge selected for field measurements of mountain radon. Research results. It has been established that at the nearby points of the mountain gorge, the prevailing direction and strength of the wind can differ significantly. The differences depend on the landscape features of the surface of the slopes of the mountain gorge. This factor determines the direction of migration of chemical elements and decay products of mountain radon. This conclusion is confirmed by direct measurement of wind speed at two points in the gorge, as well as by the results of mathematical modeling. Discussion of research results. A preliminary calculation of the passive impurity concentration field within the mountain gorge shows the most characteristic places on the slopes of the gorge from the point of view of measurements. Preliminary algorithm for choosing the location of measurements. Determination of 210Pb content in natural «traps» or sediments can determine the migration paths of «mountain» radon and estimate the magnitude of its interaction with the nature of the gorge, subject to the following assumptions: 1) certain sampling sites are not subject to significant seasonal disturbances; 2) natural and climatic conditions, especially the amount of precipitation, in the studied gorge should not have abnormal changes before sampling; 3) sampling should preferably be carried out in autumn on fallen leaves in hollows, where post-sedimentary migration is limited. Approbation of the proposed measurement technique is discussed. Conclusion. The prevailing winds in the side arms of mountain gorges can differ significantly from the wind rose observed in the main part of the gorge and are determined mainly by landscape features near the observation point. This must be taken into account when measuring and analyzing the direction of migration of chemicals in mountain gorges. Mathematical modeling of aerodynamic processes in mountainous regions is a proven method for studying physical processes in mountain gorges, as well as the processes of dispersion of volatile geochemical elements emitted by sources in mountain gorges. It is shown that at the nearby points of the mountain gorge, the wind roses can differ significantly, which is confirmed by both measurements and mathematical modeling. This dictates the need for observations and taking into account the local wind rose at each measurement point. The results of radon measurements are presented, confirming the effectiveness of the proposed technique. Resume. 1. The article presents the results of mathematical modeling of aerodynamic processes in mountain gorges and the spread of gaseous impurities in the air. 2. It has been established that wind roses at nearby points on the slopes of a mountain gorge can differ significantly, both in the results of the mathematical model and in the results of measurements from weather stations. It is shown that the agreement between the modeling and measurement results is satisfactory. 3. The research results can be useful in planning measurements of radon concentrations as a predictive map of the distribution of mountain radon concentrations.

Моделирование процесса вибрационного обезвоживания угольной мелочи в технологических системах гидромеханизированных шахт

Introduction. The article discusses the issues of modeling various variants of flow line tunneling technologies based on an analysis of the structure of the tunneling cycle. Materials and methods. As research methods, the authors used: analysis of the structures of the tunneling cycle of continuous mining technologies, mathematical modeling of processes and operations, and industrial testing of options for implementing the proposed technological tunneling systems. Results. Obtained during the implementation of a mathematical model of a high-frequency, low-amplitude process of vibration dehydration of mineral raw materials of small fractions, using fine coal (sludge) as an example, made it possible to determine the optimal parameters of the kinematics of the working body in accordance with the parameters of the source material. Innovative solutions have been proposed for the design of vibration dewatering devices with mechanical and hydraulic high-frequency flexibly adjustable drives. Conclusion. The article summarizes the results of research in the field of modeling and optimization of fine coal dewatering parameters in technological systems of mining enterprises. Options for implementing innovative technical devices for high-frequency, low-amplitude effects on the source material are proposed to increase the efficiency of dehydration. Resume. An analysis of existing means and methods for dewatering rock mass has shown that in relation to the technology of hydromechanized coal mining with its underground dewatering and recycling of process water, the most promising is the use of high-frequency vibration dewatering devices with a mechanical or hydraulic drive. The study of the kinematics of movement of the dewatered rock mass made it possible to determine the optimal values of vibration parameters

Mathematical model of the hydrocarbon displacement process by water in zonally heterogeneous deformable reservoirs

The problem of displacing hydrocarbon systems injected into the reservoir by water in zonally heterogeneous collectors is considered. In this case, the reservoir of circular shape, developed by a single central well, is assumed to consist of two zones with different reservoir and rheological properties. The solution to the problem is obtained, taking into account the incomplete displacement, compressibility of water, and PVT properties of the hydrocarbon system—phase transformation, mass exchange between phases of the hydrocarbon system. Based on the binary representation of the complex hydrocarbon system, a solution to the considered problem is obtained, and an algorithm for calculating the main indicators of the hydrocarbon displacement process to the well is proposed. This algorithm accounts for the fact that the reservoir has different reservoir and rheological characteristics in the near-wellbore region (internal zone) and the distant part of the reservoir (i.e., in the external zone). Keywords: Displacement; Gas-condensate mixture; Volatile oil; Heterogeneous reservoir; Water flooding; Binary model; Unsteady-state flow.

NONLINEAR MATHEMATICAL MODEL OF DEMAND – SUPPLY

Currently, mathematical modeling is more and more persistently invading the economy. The possibility of using mathematical modeling is related to the existence of stable trends that characterize many economic processes. The importance of modeling as a research method is determined by the fact that the model is a conceptual tool focused on the analysis of the researched processes and their forecasting. The need to master mathematical modeling of economic processes as a method of analysis is not limited to purely practical needs: mastering this method contributes to the formation of non-linear thinking. Thus, in addition to solving purely practical problems, the use of this method has a great ideological significance. The paper examines two models of supply and demand: linear and nonlinear. For each model, goals and mathematical formulations of the models are formulated. Methods for analyzing the coefficients of two models are proposed.

Automation of the Formation of a Mathematical Formulation of Kinetics for Multistage Chemical Reactions and Numerical Solution to a Direct Problem

Introduction. The basis for research, analysis and mathematical optimization of any chemical process is an adequate mathematical model that takes into account the kinetics of the object. Kinetic analysis is a challenge in chemical technology, since it allows for optimizing synthesis processes and predicting their efficiency. Numerous chemical processes involve several stage reactions. For successful design and optimization, a mathematical model that describes each stage is needed. Creating such a model manually can be time-consuming and costly, since it requires processing a large amount of information. The modern level of automation makes it possible to accelerate the obtaining of a mathematical formulation of the kinetics of multistage reactions. In this case, working with data is greatly simplified, and the probability of making mistakes is reduced. The resulting mathematical model can be applied for further analysis and optimization of the process. The paper considers the industrial reaction of catalytic reforming of gasoline, which occupies an important place in the modern scheme of oil refining, since it is a source of high-octane components of commercial gasolines and individual aromatic hydrocarbons. This process is characterized by the participation of a large number (up to 300) of various hydrocarbons, a change in the number of moles, and non-isothermality in it. Mathematical modeling of such processes involves detailing the stages to the required level. The detailing of up to 173 stages is considered. In this setting, automation of the formation of a mathematical formulation of kinetics for catalytic reforming of gasoline has not been carried out before. Therefore, the presented work aimed at implementing effective numerical methods and algorithms for automating the building of a mathematical model taking into account kinetics, thermodynamics, and changes in the number of moles.Materials and Methods. The mathematical formulation of the kinetics of multistage reactions was developed on the basis of the mass action law. The kinetic parameters values were taken from literary sources. The direct kinetics problem was solved using algorithms: the Gear method, the Runge-Kutta method of the 4th order, and the scipy.odeint() method of the Python language. The automation concept was implemented using the IDEF0 methodology. The software was written in the Python programming language.Results. A new software was created to automate the process of forming a mathematical model, taking into account the kinetics, thermodynamics, and the volume of the reaction mixture. The program results were presented by the example of catalytic reforming of gasoline. The model implemented the possibility of taking into account the intermediate heating of the mixture in the reactor cascade. Numerical values of temperature changes corresponding to industrial data were obtained.Discussion and Conclusion. The results obtained through modeling chemical transformations in the cascade of gasoline catalytic reforming reactors confirmed the exothermic nature of the reaction. The developed software product provides displaying changes in the concentrations of reactants, as well as temperature variations in the reactor, and it can be used in scientific research organizations for the analysis of multistage catalytic processes. The results of the reaction kinetics modeling will be used in the subsequent optimization of the process conditions in production.

Computer 3D Modeling of the Process of Wastewater Treatment in a Settling Tank

Purpose. Settling tanks are widely used in technological schemes of wastewater treatment. As a rule, horizontal settling tanks are used in practice. However, to increase the efficiency of water treatment in a horizontal settling tank, additional elements can be used, which are placed in different places in the structure. When reconstructing such structures or designing them for new facilities, it is necessary to assess the efficiency of the settling tanks under different operating conditions. To solve this problem, mathematical models play an important role, including conducting physical experiments, which requires considerable time to obtain results. In the case of mathematical modeling of the water treatment process in a sedimentation tank, two problems need to be solved: the hydrodynamic problem and the mass transfer problem. The main purpose of this paper is to develop a 3DCFD model for calculating the impurity concentration field in a sedimentation tank. Methodology. A dynamic multifactorial numerical model was developed for computer calculation of the wastewater treatment process in a settling tank. The hydrodynamics of the wastewater flow in the settling tank was calculated on the basis of a three-dimensional equation for the velocity potential (potential motion model). The calculation of the impurity concentration field in the settling tank was determined by numerical integration of the three-dimensional mass transfer equation. This equation takes into account the non-uniformity of the flow velocity field in the settling tank, the diffusion process, and the gravitational deposition of the impurity in the structure. The numerical integration of the modeled equations was carried out by using finite-difference splitting schemes. Findings. The software implementation of the constructed mathematical model was carried out. The results of a computational experiment to study the process of wastewater treatment in a settling tank with visualization of the calculation results are presented. Originality. A multifactorial three-dimensional CFD model has been developed that allows to quickly assess the efficiency of the settling tank. Practical value. The proposed CFD model can be used in calculations in the design of wastewater treatment systems or to determine the efficiency of wastewater treatment under new operating conditions.

Modeling the dynamic semantics of a computer program

In this paper, two approaches are used to construct the dynamic semantics of computer programs: the first is the representation of mathematical models of computational processes generated by a computer program in the form of a discrete transformer; the second is the representation of the mathematical model of a computer program in terms of functional grammars. The study focuses on the computational process generated by a computer program. Application of the discrete transducers concept to describe computational processes allows us to obtain a mathematical model or dynamic semantics of a program, which is a composition of two discrete systems B and C. In this case, component A is a control component (a model of the program block diagram), and component B is a model of the program memory. The decomposition of a program into two components B and B is convenient when studying the process of modelling a computational process. This is due to the fact that the prospects of searching for invariants in critical nodes of a programme for mathematical proof of correctness of a computer programme are opened. The comparative analysis of the two approaches leads to the theorem on the regularity of the representation of a function generated by a programme within the framework of functional grammar.

Modelling and analysis of the biodenitrification process

Massive and excessive use of nitrogen fertilizers and sustained irrigation have been widely practiced in recent years. This strategy leads to a large transfer of nitrates to groundwater, leading to a major environmental problem of nitrate contamination in water, intended for drinking water consumption. One of the most effective solutions is the degradation of nitrites and nitrates, into gaseous nitrogen, using the heterotrophic bacteria during the denitrification process. In this paper, we present and study a mathematical model of the biodenitrification process taking into account the fixed and mobile bacteria. This process is modeled by a system of ordinary differential equations and requires the success of bacteria to colonize the reactor. We study the existence and the asymptotic behaviour of the solution. We show the existence of a value of the injected carbon concentration from which we ensure the success of the biodenitrification process and we propose a heuristic algorithm which serves to control the biodenitrification process over time. Finally, we present some numerical simulations in to support the theoretical results.

Mathematical modeling of oil solid-liquid extraction from mamey sapote (Pouteria sapota) seeds with hexane

Mamey sapote (Pouteria sapota) seeds oil has potential as human consumption oil due to its high content of oleic acid. In order to design a solid-liquid extraction sustainable process, (hexane)-(oil)-(Pouteria sapota seeds) extractive system was modeled for different extraction modes applying novel mathematical relations. One equilibrium stage solid-liquid extractions (SLE) and dynamic SLE with continuous solvent flow through a particles fixed bed were experimentally performed in order to estimate the extractive system thermodynamic properties. The results for thermodynamic distribution constant, specific retained solution and oil diffusivity in underflow (seeds-extract mixture) were 0.24 g/g, 1.58 g/g and 8.68 × 10−11 m2/s (at 30 °C) respectively. These properties, applied in the mathematical models of counter-current equilibrium multistage and dynamic SLE processes, reproduced experimental behavior with 4.53–16.7 average difference %.

ARRA TISHLARIDAN PAXTA TOLASINI YECHIB OLISH JARAYONI PARAMETRLARINI ILMIY ASOSLASH

This paper scrutinizes the process of extracting or separating fiber from seed, which is viewed as one of the milestones in cotton-ginning plants, reviews a series of implemented experiments, presents results, and offers solutions. Increasing demand for cotton produce across the globe leads to large-scale research work to improve the technique and technology of primary cotton processing and to establish their scientific basis in major cotton-growing economies. In this regard, it is becoming important, among other things, to develop mathematical models of processes, boost the productivity of cotton processing machines by means of optimization methods, create efficient working bodies, preserve natural quality indicators of the cotton fiber obtained through improved operational characteristics of machines, and reduce production costs. At the same time, the issues of the development of resource efficient designs of working bodies that won’t negatively affect the quality of products and the justification of the parameters of the sewing machine seem relevant for all cotton-growing economies.

Analyzing production optimization models across various carbon pricing policies

As the EU and the US successively proposed the carbon border adjustment mechanism (CBAM) and the Clean Competition Act (CCA), importers are now required to declare the carbon emissions of their products. Particularly, wheel companies face increasing time pressure with the gradual development of implementation paths for net-zero emissions and implementing carbon pricing policies in various countries. This research uses the production data of Taiwan wheel companies to establish a mathematical model for the activity-based cost optimisation decision-making process. It explores the impact of different carbon emission costs on a company’s product mix with pricing decisions. Further, the research proposes five mathematical programmes for single-period cost optimisation decision-making under carbon pricing. These models provide some detailed view, such as changes in profits and product mix caused by high carbon taxes or other changes for each resource. Furthermore, the production cost optimal model proposed in this research can be used by enterprises to fulfill their social responsibilities while ensuring profitability and serve as a reference for making net-zero transformation decisions.

MATHEMATICAL MODELS OF CHEMICAL PROCESSES WITH VARIABLE CHARACTERISTICS

The article discusses the construction of mathematical models of complex chemical processes when different factors act on the result of a chemical reaction at different points in time. The method of least squares is used. This material can be used to teach university students in the disciplines of Mathematical Statistics, Numerical Methods, General Technologies of Food Production, Food Chemistry, etc. Applications from the analysis of experimental research models and laboratory testing of innovative technologies for the development of different types of galusa can be supported by incoming students in the chemical engineering program. Examples from the analysis of experimental research models of laboratory tests of new acid components for a carbonate collector can be applied in the training of students in the Chemical Engineering educational program.

Математическое моделирование при управлении системами противопожарной защиты объектов

The problem of reliability of fire protection systems at the production stage and at the design stage is considered. Mathematical modeling of processes aimed at ensuring the safe evacuation of people in case of fire, taking into account the coefficient of compliance of the technical solution used in the methodology for calculating the amount of individual fire risk. The structural constructions of fire protection systems are analyzed with a demonstration of methods for assessing the reliability and structural significance of various structural constructions of fire protection systems on the basis and example of statistical data. To determine the coefficient of compliance of the fire alarm system, in the form of probabilities of effective operation of the fire alarm system, the following parameters are taken into account: the probability of effective operation of the fire alarm system, smoke protection system, and emergency warning system. The performance statistics of the SOUE for the period from 2013 to 2021 are graphically shown. It is shown that the resulting probability of effective operation of the SPS is determined by the structural design, and the corresponding formulas and examples are given. The dependences of the efficiency of the functioning of the SPZ on the efficiency of the SOUE for various structural arrangements have been constructed. Dependencies of the performance of the SDR were constructed based on statistical data. Analytical and graphical dependences of the reliability of the SOUE on the reliability of the SPS for various structural constructions are derived. Depending on the task being solved, the reliability (as well as efficiency) of the fire protection system can be controlled, while the reliability of the fire safety protection system can be determined depending on the required fire risk. Thus, one of the most important measures that minimizes individual risk should be considered to be increasing the efficiency of each of the subsystems included in the SDR, including the SOUE, as well as optimizing the structural design of the SDR.

Mathematical Modeling of Thermal Processing for an Al-Si Alloy

This paper presents the realization of a mathematical model of a heat treatment process of an Al-Si alloy, with the help of which the values of some mechanical properties are preached. The method of elaborating the mathematical model is a statistical method, namely the method of the active experiment.With the help of the MATLAB software and based on the elaborated mathematical model, we created a graphic interface that allows the prediction of the values of three mechanical properties depending on the thermal processing conditions of the studied alloy.

DEVELOPMENT OF ENERGY-EFFICIENT AIR DEDUSTING

Among the sources of internal pollution one can point to dust, breath products of passengers, various emissions from finishing materials of trains and metro stations, etc. This problem is usually solved in two ways: wet cleaning of tunnels and stations and using powerful tunnel vacuum cleaners. Both of these methods have common disadvantages. Firstly, dust removal can only be carried out at night, when there are no trains on the line. Secondly, the dust removal installations themselves are expensive and are effective only at low speeds of dust movement through the tunnel. All metros in the world have a problem with dust removal. Research methods include analytical and computational studies of air distribution in the metro ventilation network using mathematical modeling methods using graph theory and flow algorithms, as well as experimental studies of air flow parameters in natural conditions of the Almaty metro.This article proposes an innovative method for dust removal of tunnel air based on the use of piston action of trains and labyrinth filters installed in station ventilation joints. The parameters of the calculation model of the metro line are substantiated on the basis of a decomposition approach to the mathematical modeling of aerodynamic processes by moving from a linear model of the metro line to a periodic open-loop model. Computational aerodynamics methods were used to determine the patterns of changes in air speed through the ventilation vent and the available pressure drop depending on the speed and mode of trains passing through the station.

Numerical analysis of temperature effect on CO2 storage capacity and CH4 production capacity during the CO2-ECBM process

Temperature is a key factor affecting the CO2 injection efficiency and CH4 production efficiency, so it is of great significance to clarify the evolution diagram of temperature change during the CO2-ECBM process. In this study, the crushed soft coal seam with low permeability in Huainan coalfield was taken as the research object. Firstly, the fully coupled mathematical models of CO2-ECBM process were constructed, and the influence of temperature field on CO2-ECBM process was emphasized. Secondly, the evolution law of CO2-ECBM process at different time was revealed. Then, the effects of injection temperature and initial temperature on the CO2-ECBM process were analyzed. Finally, the evolution mechanism of reservoir temperature during the CO2-ECBM process was illustrated, and the effect of temperature on temperature field of CO2-ECBM process was further illustrated. The results show that the longer the displacement time, the CH4 concentration gradually decreased, and the CO2 concentration gradually increased. The area where the temperature changes are more concentrated focuses on the vicinity of CO2 injection well, which are mainly determined by the dominant reaction of adsorption and desorption reactions of binary gas. Reservoir temperature and permeability are positively and negatively correlated with injection temperature, respectively. Both CO2 storage capacity and CH4 production capacity increase with the increase of injection temperature, and the sensitivity of CO2 storage capacity to temperature was greater than that of CH4 production capacity. With the increase of displacement time and injection volume, the effective displacement radius increases, but the increment of influencing radius gradually decreases. The lower the initial temperature of coal reservoir, the larger the influencing radius, and which increases with the increase of displacement time. CO2 storage capacity and CH4 production capacity decrease with the increase of initial temperature. Young's modulus and Poisson's ratio can effectively change the reservoir strength. CO2 storage capacity and CH4 production capacity decrease with the increase of Young's modulus, and increase with the increase of Poisson's ratio. This study can support the engineering landing of CO2-ECBM technology in crushed soft coal seam with low permeability, and promote the realization of China's “Dual Carbon” strategic goal.

Formation and Growth of Co-Culture Tumour Spheroids: New Compartment-Based Mathematical Models and Experiments.

Co-culture tumour spheroid experiments are routinely performed to investigate cancer progression and test anti-cancer therapies. Therefore, methods to quantitatively characterise and interpret co-culture spheroid growth are of great interest. However, co-culture spheroid growth is complex. Multiple biological processes occur on overlapping timescales and different cell types within the spheroid may have different characteristics, such as differing proliferation rates or responses to nutrient availability. At present there is no standard, widely-accepted mathematical model of such complex spatio-temporal growth processes. Typical approaches to analyse these experiments focus on the late-time temporal evolution of spheroid size and overlook early-time spheroid formation, spheroid structure and geometry. Here, using a range of ordinary differential equation-based mathematical models and parameter estimation, we interpret new co-culture experimental data. We provide new biological insights about spheroid formation, growth, and structure. As part of this analysis we connect Greenspan's seminal mathematical model to co-culture data for the first time. Furthermore, we generalise a class of compartment-based spheroid mathematical models that have previously been restricted to one population so they can be applied to multiple populations. As special cases of the general model, we explore multiple natural two population extensions to Greenspan's seminal model and reveal biological mechanisms that can describe the internal dynamics of growing co-culture spheroids and those that cannot. This mathematical and statistical modelling-based framework is well-suited to analyse spheroids grown with multiple different cell types and the new class of mathematical models provide opportunities for further mathematical and biological insights.