Dairy Science & Technology · Postgraduate & Research · also Advances in Dairy Engineering, Dairy Process Engineering (PG)
Graduate-level transport phenomena, process modelling, design and simulation for dairy processes, as examined in ICAR AICE subject 38.
In the exams: ICAR AICE subject 38 (all units); ASRB NET Dairy Engineering
Advanced dairy engineering applies transport phenomena, modelling, simulation, design codes and control theory to the problems of modern dairy plants. Where undergraduate dairy engineering teaches how pasteurisers, evaporators, dryers and membrane plants work, this subject asks how to model them quantitatively: how heat, mass and momentum move through the product; how much lethality a heat treatment delivers to the fastest particle; how fouling builds up and how cleaning can be optimised; how membranes, spray dryers and freezing can be predicted; how process simulators, design codes and advanced control are used; and how energy and water use can be minimised systematically.
The manual covers ten units at postgraduate and research level: transport phenomena in dairy processes; modelling of heat treatment and lethality; fouling models and cleaning optimisation; membrane process modelling; spray dryer modelling and computational fluid dynamics; freezing and thawing models; process simulation software; pressure vessel and heat exchanger design codes; advanced control and machine learning in dairy plants; and sustainability engineering through energy and water pinch analysis. It builds on fluid flow, heat transfer and thermodynamics, and on the subjects on dairy processing equipment, evaporation and drying, refrigeration and instrumentation, without repeating them.
Doctoral fellowship and research-service examinations in dairy engineering ask candidates to derive and apply models, interpret simulation and design results, and choose methods for real problems. Design codes and regulations are named as examples from several jurisdictions; the principles are common, but any design must follow the code that applies where the equipment is built and used. Commercial software is described by type only.
Transport phenomena treat the movement of momentum, heat and mass as parallel processes governed by similar laws. Newton's law of viscosity relates shear stress to velocity gradient through viscosity; Fourier's law relates heat flux to temperature gradient through thermal conductivity; and Fick's law relates mass flux to concentration gradient through diffusivity. The corresponding diffusivities, kinematic viscosity ν, thermal diffusivity α and mass diffusivity D, all have units of m²/s, and their ratios define the Prandtl number Pr = ν/α, the Schmidt number Sc = ν/D and the Lewis number Le = α/D.
Near a wall, momentum, heat and mass transfer occur across boundary layers whose thicknesses depend on these ratios: for milk, with a Prandtl number of roughly 20 to 30 at room temperature, the thermal boundary layer is much thinner than the velocity boundary layer, and for most solutes in liquids, with Schmidt numbers of hundreds to thousands, the concentration boundary layer is thinner still. In turbulent flow, analogies link the transfer coefficients: the Reynolds analogy, St = f/2 for Pr = 1, where St is the Stanton number and f the Fanning friction factor, and the Chilton–Colburn analogy, j_H = St Pr^(2/3) = f/2 and j_D = St_m Sc^(2/3) = f/2, which extends to other Prandtl and Schmidt numbers. These analogies allow a heat-transfer coefficient to be estimated from a pressure-drop measurement, or a mass-transfer coefficient from heat-transfer data.
Interfacial mass transfer is described by models of increasing realism: the film theory, in which transfer occurs by steady diffusion across a stagnant film so that k = D/δ; the penetration theory of Higbie, in which fluid elements contact the interface for a fixed time and k ∝ √(D/t); and the surface renewal theory of Danckwerts, with random renewal of surface elements. They predict different dependence on diffusivity, which experiments can distinguish.
In continuous processes, not every fluid element spends the same time in the equipment. The residence time distribution (RTD) describes this. The exit age distribution E(t) gives the fraction of fluid leaving at times between t and t + dt; its integral, the cumulative distribution F(t), gives the fraction that has left by time t. E(t) is measured by injecting a tracer pulse, such as salt detected by conductivity, at the inlet and recording its concentration at the outlet; F(t) comes from a step change. The mean residence time is the first moment of E(t) and equals V/Q for a closed vessel; the variance measures spread.
Ideal reactors bound real behaviour. In plug flow, all elements spend exactly V/Q; in a perfectly mixed vessel (continuous stirred tank), E(t) = (1/τ)exp(−t/τ), so many elements leave almost immediately. Real equipment lies between, and is modelled by tanks-in-series models, in which N equal mixed tanks reproduce the observed variance (N = τ²/σ²), or by the axial dispersion model, characterised by a Péclet number comparing convection with axial dispersion; large Péclet numbers approach plug flow.
RTD matters in dairy processing because quality and safety depend on the time–temperature history of each element. In laminar flow of a Newtonian liquid in a tube, the fastest element at the centre travels at twice the mean velocity, so the minimum residence time is half the mean; molecular diffusion broadens this into Taylor dispersion. Turbulent flow and secondary flows in coils and corrugated channels narrow the RTD. In holding tubes, scraped-surface heat exchangers, fermenters and continuous freezers, the RTD determines how many elements are under- or over-processed, and models combining the RTD with kinetics predict the distribution of lethality or quality across the product.
Many dairy processes combine flow, heat and mass transfer with changing properties. In heat exchangers, milk viscosity falls several-fold as it heats, which changes the velocity profile and heat-transfer coefficient along the channel, and the wall temperature, higher than the bulk, drives protein denaturation and fouling at the wall rather than in the bulk. Non-Newtonian products, such as concentrates and yoghurt, have flatter velocity profiles and apparent viscosities that depend on shear rate, which are handled with generalised Reynolds numbers and with numerical solution of the momentum and energy equations.
Conjugate heat transfer problems couple conduction in a solid wall with convection in the fluids on both sides, and are solved numerically when wall resistance or axial conduction matters, as in thick-walled tubes or heat exchangers with low flow. Coupled heat and mass transfer occurs in drying, evaporation and freezing, where latent heat links the two. In particulate flows, such as cheese curd in whey or fruit pieces in a dessert, particles move at different speeds from the carrier fluid and receive different heat treatments.
The governing equations of these problems, the continuity, Navier–Stokes, energy and species conservation equations, are partial differential equations solvable analytically only for simple cases. Numerical methods, finite difference, finite volume and finite element, solve them for real geometries, and computational fluid dynamics software built on these methods is now a standard engineering tool, discussed later for spray dryers. Dimensionless analysis remains essential for interpreting numerical results and for scale-up.
The lethality of a heat treatment is calculated by integrating the lethal rate over the time–temperature history of the product: F = ∫10^((T − T_ref)/z) dt. The result is the equivalent time at the reference temperature T_ref that would give the same microbial inactivation, for organisms whose resistance has the stated z value. For sterilisation of low-acid foods, the F₀ value uses T_ref = 121.1 °C and z = 10 °C; the minimum botulinum cook, designed to give twelve decimal reductions of Clostridium botulinum spores, corresponds to F₀ of about 3 minutes, based on a D₁₂₁ of about 0.2 to 0.25 minutes. Pasteurisation of milk is expressed by equivalence to standard time–temperature combinations, such as 72 °C for 15 s or 63 °C for 30 min, which are based on inactivation of the most heat-resistant non-spore-forming pathogens historically relevant to milk.
For UHT processes, Kessler introduced two dimensionless indices. The bacteriological effect B* is defined so that B* = 1 corresponds to the lethality of 135 °C for 10.1 s with z = 10.5 °C, equivalent to about nine decimal reductions of thermophilic spores; the chemical effect C* is defined so that C* = 1 corresponds to 135 °C for 30.5 s with z = 31.4 °C, equivalent to about 3% destruction of thiamine. A good UHT process achieves B* above 1 while keeping C* below 1, and because the z values differ, raising temperature and shortening time increases B* relative to C*, the high-temperature short-time principle expressed quantitatively.
Lethality must be calculated over the whole process, including heating and cooling, not just the holding section, and in continuous processes for the fastest-moving element, which receives the least lethality.
The general method, attributed to Bigelow, integrates measured or modelled temperature at the critical point numerically, by the trapezoidal or Simpson rule applied to the lethal rate at each time step. It is exact for the measured profile and is the basis of validating real processes from data-logger records. For in-container sterilisation, temperature is measured at the slowest-heating point of the container, the cold spot, which for conduction-heating foods is near the geometric centre and for convection-heating liquids lies lower, on the axis.
Formula methods predict lethality from parameters of heat penetration curves. Ball's method describes the heating curve of a container by two parameters, the heating rate index f_h, the time for the temperature difference between retort and product to fall tenfold, and the lag factor j_h, and uses tables to calculate the process time needed for a given F₀ or the F₀ delivered by a given time; variants account for broken heating curves and cooling. Such methods allow process times to be recalculated for changes in retort temperature or initial temperature without new trials.
Numerical models of heat transfer, from analytical solutions of conduction for simple shapes to finite-element and computational fluid dynamics models of containers and continuous heaters, predict temperature histories at all points, allowing the distribution of lethality and of quality changes to be calculated. They are used to design processes, to evaluate deviations, such as a temporary drop in retort temperature, and to optimise processes for quality, for example by choosing temperature and time to achieve the required F₀ with minimum nutrient loss. Process authorities review such calculations before processes for low-acid foods are filed or approved in jurisdictions that require it.
In continuous heat treatment, lethality depends on both the temperature profile and the residence time distribution. For a Newtonian liquid in laminar flow, the fastest element moves at twice the mean velocity, so holding tubes are sized for half the mean residence time unless turbulent flow is demonstrated; in turbulent flow the fastest element moves at about 1.2 times the mean. For non-Newtonian fluids with flow behaviour index n, the ratio of maximum to mean velocity is (3n + 1)/(n + 1). Temperature also varies across the tube, and the cooler regions near the centre in heating sections must be considered.
Aseptic processing of liquids containing particles, such as desserts with fruit pieces, requires the centre of the slowest-heating particle that also travels fastest to receive the required lethality. Because particle residence times and fluid-to-particle heat-transfer coefficients are difficult to measure, conservative assumptions or validated measurements with simulated particles carrying magnets or temperature indicators are used, together with biological indicators, spores immobilised in the particles, to demonstrate lethality.
Uncertainty and variability affect all lethality calculations: variability in D and z values between strains and in different media, uncertainty in temperature measurement and location, variation in initial microbial load, and variation in product properties. Stochastic models propagate these uncertainties to give distributions of outcomes, for example the probability that a pack is non-sterile, and are increasingly used in risk-based process design. Safety margins in established processes, such as the 12D concept, implicitly account for these uncertainties, which is why reducing them requires strong evidence.
Fouling, the deposition of material on heat-transfer surfaces, is one of the main operating problems of dairy heat exchangers and evaporators. It lowers the overall heat-transfer coefficient, increases pressure drop, lowers outlet temperatures, provides sites for bacterial growth and limits run times, forcing frequent cleaning. Milk fouling is classically divided into two types. Type A deposits form at about 75 to 110 °C; they are soft, voluminous and white to cream-coloured, consisting of roughly half to two thirds protein, mainly denatured β-lactoglobulin, with minerals and fat. Type B deposits form at higher temperatures, above about 110 °C, as in UHT plants; they are hard, compact and granular, consisting largely of minerals, mainly calcium phosphate, with a smaller protein fraction.
The mechanism of Type A fouling is linked to β-lactoglobulin denaturation. Native β-lactoglobulin unfolds, exposing its reactive thiol, and the unfolded molecules aggregate in the bulk and react at the wall. Kinetic models, such as those developed by de Jong and colleagues, relate the local fouling rate to the concentration of unfolded β-lactoglobulin near the wall, calculated from denaturation kinetics along the heat exchanger; they predict that deposition is greatest where the product reaches temperatures at which unfolding is rapid but aggregation in the bulk has not yet consumed the reactive protein. Calcium phosphate, whose solubility falls with temperature, deposits increasingly at higher temperatures and binds the protein layer.
Pre-holding milk at a moderate temperature to denature β-lactoglobulin before it reaches the heating section reduces deposition in that section.
Fouling is quantified as a fouling resistance R_f, the additional thermal resistance of the deposit: 1/U_fouled = 1/U_clean + R_f. Plotting R_f against time gives curves of several shapes. Linear growth, constant deposition rate, is typical of hard scales; falling-rate curves show decreasing deposition as conditions change; and asymptotic curves, described by the Kern–Seaton model R_f = R_f∞(1 − exp(−t/τ)), arise when deposition is balanced by removal, as in some soft deposits under high shear. Many milk fouling curves show an induction period, during which little resistance develops while the first layer forms, followed by rapid growth.
Kern and Seaton's model treats the net rate as deposition minus removal, with removal proportional to the deposit thickness and the wall shear stress. Mechanistic models for milk combine protein reaction kinetics, mass transfer to the wall and surface reaction.
Fouling models are used in design to choose surface area allowances, flow velocities and temperature programmes, and in operation to predict run lengths and schedule cleaning. Measured indicators of fouling, such as rising pressure drop, increasing heating-medium temperature needed to maintain product temperature, and falling overall heat-transfer coefficient calculated from temperatures and flows, are monitored to decide when to clean.
Cleaning-in-place removes deposits by combining four factors, often represented as Sinner's circle: chemical action, temperature, mechanical action from flow and time; reducing one requires increasing others. Typical dairy cleaning uses an alkaline wash, commonly sodium hydroxide at around 0.5 to 2% at elevated temperature, to swell and remove proteinaceous deposits, followed by an acid wash, such as nitric or phosphoric acid, to dissolve mineral deposits, with rinses between and after; single-phase formulated detergents with chelants and surfactants are also used. The removal of protein deposits by alkali proceeds through phases studied by Bird and Fryer and others: a swelling phase in which the deposit absorbs alkali and becomes a gel, a uniform removal phase at roughly constant rate controlled by shear and diffusion, and a decay phase in which thin, isolated patches are removed slowly. Excessively concentrated alkali can slow cleaning by forming a less removable gel, so concentration has an optimum.
Optimising cleaning involves several decisions. The run time between cleanings balances production lost to cleaning against declining performance and increasing deposit: average productivity over a cycle of production time t and cleaning time t_c is maximised at an optimal t that depends on how quickly performance falls. Cleaning programmes are optimised by monitoring the end of each phase, for example by conductivity, turbidity or deposit sensors, rather than using fixed times; by recovering and reusing alkali and acid solutions, with membrane filtration to remove soil from used caustic; by using enzymes for difficult deposits; and by minimising rinse water through phase separation based on conductivity. Cleaning must always be verified, by visual inspection, ATP and microbiological swabs, and monitoring of time, temperature, concentration and flow against validated limits.
In pressure-driven membrane processes, the permeate flux J is driven by the transmembrane pressure and opposed by resistances. The resistance-in-series model writes J = TMP/(μ(R_m + R_c + R_f)), where μ is permeate viscosity, R_m the clean membrane resistance, R_c the resistance of the concentration polarisation layer or cake, and R_f that of fouling within or on the membrane. For reverse osmosis and nanofiltration, where small solutes are retained, the osmotic pressure difference across the membrane opposes flow: J = (ΔP − σΔπ)/(μR_m), where σ is the reflection coefficient. Osmotic pressure can be estimated by the van 't Hoff relation π = iCRT for dilute solutions; milk, with an osmotic pressure of roughly 0.7 MPa, requires reverse osmosis pressures several times higher to achieve useful fluxes at high concentration.
In ultrafiltration and microfiltration of milk, flux increases with pressure at first, then levels off at a limiting flux independent of pressure, because retained proteins accumulate at the membrane in a polarised boundary layer. The film or gel polarisation model describes this: at steady state, convective transport of solute toward the membrane equals back-diffusion, giving J = k ln((C_w − C_p)/(C_b − C_p)), where k is the mass-transfer coefficient, C_w the wall concentration, C_b the bulk concentration and C_p the permeate concentration; when C_w reaches a limiting gel concentration, flux depends only on k and C_b, falling logarithmically as the retentate is concentrated. Mass-transfer coefficients come from correlations, such as the Lévêque solution for laminar flow, Sh = 1.62(Re Sc d_h/L)^(1/3), and turbulent correlations similar to Dittus–Boelter.
The critical flux concept states that below a certain flux, fouling is negligible; operating near but below it is a common design strategy for long, stable runs.
Flux decline over time at constant pressure is analysed with blocking models proposed by Hermia: complete pore blocking, in which each particle seals a pore; standard blocking, in which material deposits inside pores and narrows them; intermediate blocking, combining sealing and superposition; and cake filtration, in which retained material forms a growing layer. Each predicts a characteristic relationship between filtration time and volume, and fitting these relationships identifies the dominant mechanism, often changing from pore blocking early to cake formation later.
The progress of a batch or continuous concentration is described by mass balances. The volume concentration factor VCF is the ratio of feed volume to retentate volume. The observed rejection of a solute is R = 1 − C_p/C_b. For a solute with constant rejection R in a batch concentration, the retentate concentration rises as C_R = C_0 × VCF^R; fully retained solutes (R = 1) concentrate in direct proportion to VCF, and freely passing solutes (R = 0) stay at the feed concentration. For example, ultrafiltration of skim milk to a VCF of 4 raises protein, with R near 1, about fourfold, while lactose, with R near zero, stays at about its original concentration in the retentate liquid.
Diafiltration washes permeable solutes out of the retentate by adding water. In continuous, constant-volume diafiltration, the concentration of a solute of rejection R falls as C = C_0 exp(−N(1 − R)), where N is the number of diavolumes, the volume of water added divided by the retentate volume; for a freely permeable solute, five diavolumes reduce its concentration to about 0.7% of the starting value. This is how milk protein isolates and whey protein isolates are purified of lactose and minerals.
Membrane modules for dairy applications include spiral-wound polymeric modules, compact and cheap, used for reverse osmosis, nanofiltration and ultrafiltration of milk and whey; tubular polymeric and ceramic modules, used for viscous and particulate streams; hollow-fibre modules; and ceramic multichannel elements for microfiltration. Ceramic microfiltration at about 1.4 µm pore size removes bacteria and spores from skim milk, extending shelf life, while about 0.1 µm separates casein micelles from whey proteins. In ceramic microfiltration, the uniform transmembrane pressure concept, which circulates permeate co-currently to keep the pressure difference constant along the long module, avoids high local flux and fouling at the inlet.
Plant design combines flux models, mass balances and economics. Staged designs, with several loops in series operating at increasing concentrations, approach the efficiency of batch operation in continuous plants. Temperature affects flux through viscosity, microbial growth and protein behaviour; dairy membrane plants commonly operate either cold, around 10 to 15 °C, to limit bacterial growth during long runs, or warm, around 50 °C, for higher flux with shorter runs. Pumping energy, mainly for cross-flow circulation, dominates operating costs in ultrafiltration and microfiltration, while pressure energy dominates in reverse osmosis.
Models are used to size membrane area for a required throughput, to predict flux decline and cleaning frequency, to choose operating pressures and velocities, and to design diafiltration schedules that achieve target purity with minimum water. Plant data are fed back to refine model parameters, and cleaning regimes are chosen for membrane material compatibility, since polymeric membranes have limited tolerance of high pH, temperature and chlorine compared with ceramics.
A spray dryer can be modelled at three levels: whole-dryer heat and mass balances, single-droplet drying kinetics, and the coupled flow of air and particles in the chamber. The heat and mass balance relates inlet and outlet air temperatures and humidities, feed and powder flows and compositions, and heat losses; from it the evaporation rate, air flow, and thermal efficiency follow. A simple thermal efficiency is η = (T_in − T_out)/(T_in − T_ambient); for inlet air at 200 °C, outlet at 90 °C and ambient at 20 °C, it is 110/180, about 61%. Higher inlet temperatures and lower outlet temperatures raise efficiency, within the limits set by product damage, stickiness and the moisture specification.
A drying droplet passes through stages. After brief initial heating, a droplet of milk concentrate dries at a nearly constant rate while its surface remains saturated, with the surface near the wet-bulb temperature of the air, so the product stays relatively cool even in very hot air. As solids accumulate at the surface, a skin or crust forms, the rate falls, and the particle temperature rises toward the air temperature; in this falling-rate period, internal diffusion of water through the increasingly concentrated, viscous material controls drying, and the particle may inflate, collapse or form a hollow shell, depending on conditions and composition.
Models of droplet drying range from the characteristic drying curve approach, which scales the drying rate by the constant-rate value as a function of normalised moisture, to the reaction engineering approach developed by Chen and colleagues, which treats evaporation as an activation process with an apparent activation energy depending on moisture content, and to detailed models solving diffusion and heat conduction inside the droplet with moving boundaries. Single-droplet experiments, with droplets suspended on filaments or levitated, provide data to fit and validate these models.
Computational fluid dynamics (CFD) solves the conservation equations of mass, momentum, energy and species numerically over a mesh representing the dryer. For spray drying, the air is usually modelled as a continuous phase in an Eulerian framework, with turbulence represented by Reynolds-averaged models such as k–ε or k–ω variants, while droplets and particles are tracked individually in a Lagrangian framework as a discrete phase, each parcel representing many particles of a given size. The phases are coupled: droplets exchange momentum, heat and mass with the air, which changes the air's temperature and humidity, and the particles' drying is calculated along their trajectories with a droplet drying model.
Input data include the atomiser's droplet size distribution, from correlations or measurements, initial velocities and spray angle; the inlet air flow, temperature, humidity and swirl; chamber geometry; and properties of the feed. Outputs include air flow patterns, temperature and humidity fields, particle trajectories, residence times, final moisture and temperature histories, and regions where particles hit walls.
Wall deposition is a key application. Particles hitting a wall stick if they are sticky, which depends on how far their surface temperature is above the glass transition temperature, commonly expressed by a sticky-point criterion such as T − Tg greater than about 10 to 20 K; CFD combined with glass transition models predicts where deposits will form, which guides air-distributor design, swirl and outlet conditions. Limitations include the computational cost of modelling agglomeration, particle collisions and fines return, uncertainty in drying kinetics, and the simplifications of turbulence models; validation with temperature, humidity and velocity measurements in real dryers remains essential.
Models are used across the life of a dryer. In design, they help choose chamber dimensions, inlet configurations and atomiser type for a required capacity and powder specification, and evaluate new concepts such as tall-form or multi-stage dryers. In scale-up, CFD complements pilot trials by showing how flow patterns and residence times change with size. In operation, simpler models based on heat and mass balances, calibrated to plant data, predict powder moisture from inlet and outlet conditions and feed rates, and serve as soft sensors in control systems that adjust outlet temperature or feed rate to keep moisture at its target despite changes in ambient humidity and feed solids.
Models also link drying to product quality. Combining particle temperature–moisture histories with kinetics of protein denaturation, Maillard reaction, enzyme inactivation and bacterial inactivation predicts functional properties and survival of probiotics. Particle formation models predict morphology, density and surface composition; for example, fat and proteins tend to be over-represented at particle surfaces relative to their bulk content, because of differences in diffusion during drying, which affects wettability and oxidative stability.
Energy and environmental analysis uses dryer models to evaluate heat recovery from exhaust air, the effects of higher feed solids from improved evaporation, and alternatives such as heat pumps supplying inlet air heat. Because dryers are often the largest single energy user in powder plants, even modest efficiency improvements identified by modelling have significant economic and carbon benefits.
Freezing removes sensible and latent heat from a product. A typical freezing curve shows precooling to the initial freezing point, sometimes with brief supercooling below it before nucleation; a phase-change period during which most of the water freezes and temperature falls slowly as the remaining solution becomes more concentrated; and tempering to the final storage temperature. Because solutes depress the freezing point and become concentrated as ice forms, foods do not freeze at a single temperature: milk begins to freeze at about −0.52 to −0.55 °C, ice cream mix at about −2 to −3 °C depending on sugar content, and the unfrozen fraction decreases progressively as temperature falls.
The fraction of water frozen at a temperature below the initial freezing point can be estimated from freezing point depression: assuming ideal behaviour, the unfrozen water fraction is approximately the ratio of the initial freezing point depression to the depression at the temperature considered. For an ice cream mix with an initial freezing point of −2.5 °C, at −10 °C roughly 2.5/10 = 25% of the water remains unfrozen and about 75% is ice, and at −20 °C about 87.5% is ice. This explains why ice cream is spoonable at freezer temperatures and why storage temperature strongly affects texture.
Freezing rate determines ice crystal size: rapid freezing produces many small crystals, slow freezing fewer, larger ones. In ice cream, the dynamic freezing in a scraped-surface freezer produces small crystals, about 10 to 20 µm on average, and the hardening stage should be fast so that remaining water freezes onto existing crystals; during storage, temperature fluctuations cause recrystallisation and coarse texture. In butter, cheese and cream, freezing affects fat crystallisation and emulsion stability, and slow freezing can damage cheese texture.
Plank's equation gives a simple estimate of the freezing time for a product initially at its freezing point: t_f = (ρL/(T_f − T_a)) × (P a/h + R a²/k), where ρ is density, L latent heat per unit mass of the product, T_f the initial freezing point, T_a the cooling medium temperature, a the characteristic dimension (thickness of a slab or diameter of a cylinder or sphere), h the surface heat-transfer coefficient, k the thermal conductivity of the frozen product, and P and R shape factors: P = 1/2 and R = 1/8 for an infinite slab, 1/4 and 1/16 for an infinite cylinder, and 1/6 and 1/24 for a sphere.
The equation shows the structure of the problem: freezing time is inversely proportional to the temperature driving force; the first term represents surface resistance and is proportional to size; the second represents internal conduction resistance and is proportional to size squared, so for large blocks, such as cartons of butter, conduction dominates and doubling thickness roughly quadruples freezing time. For example, a slab of butter-like product 0.05 m thick with ρ = 950 kg/m³, L = 50 kJ/kg (reflecting its modest water content), T_f − T_a = 25 K, h = 15 W/m² K and k = 0.2 W/m K gives t_f ≈ (950 × 50,000/25) × (0.5 × 0.05/15 + 0.125 × 0.0025/0.2) ≈ 1.9 × 10⁶ × (0.00167 + 0.00156) ≈ 6,100 s, about 1.7 hours, of which surface and internal resistances contribute almost equally.
Plank's equation ignores precooling and tempering and assumes freezing at a single temperature, so it underestimates real freezing times. Refinements, such as those of Cleland and Earle and of Pham, add terms for sensible heat and use effective latent heats and shape factors for real geometries, and numerical methods using enthalpy formulations or apparent specific heat solve the full transient problem with temperature-dependent properties.
Thawing is generally slower than freezing under comparable temperature differences, for physical reasons. In freezing, the frozen outer layer, through which heat must pass, is ice, with a thermal conductivity of about 2.2 W/m K; in thawing, the outer layer is liquid water, with a conductivity of about 0.6 W/m K, roughly four times lower. Thawing temperature differences are also limited, because the surface must not get warm enough to allow microbial growth or quality damage while the centre is still frozen. Dairy products such as frozen cream, butter, curd and concentrated milk are thawed in controlled rooms, with water or with microwave and radio-frequency tempering, the latter giving volumetric heating that avoids the conduction limitation but risks uneven heating and runaway, because liquid water absorbs microwave energy more than ice.
Frozen storage quality depends on temperature and its stability. Recrystallisation of ice, sublimation from surfaces causing freezer burn, lactose crystallisation in ice cream and dried products, and slow chemical changes such as lipid oxidation all continue during storage, more rapidly at higher and fluctuating temperatures. Models of recrystallisation, based on the kinetics of crystal growth and on the glass transition of the freeze-concentrated phase, show that storage near or below Tg′ greatly slows these changes, while storage several degrees above Tg′, common for ice cream, allows them.
Freezing and thawing models are used to design freezers and hardening tunnels, to size refrigeration capacity, to set the time products must spend in blast freezers, to assess the effects of cold-chain interruptions on product temperature, and to evaluate energy use. Their accuracy depends on property data: density, specific heat, thermal conductivity and enthalpy as functions of temperature, which are often estimated from composition using predictive models when measured data are unavailable.
Process simulation software represents a plant as a flowsheet of unit operations connected by streams, and calculates mass and energy balances, equipment sizes, utilities and often costs. Steady-state simulators solve the balances for a plant at constant conditions; dynamic simulators solve differential equations for the plant's behaviour over time, used for start-up, shutdown, control design and operator training. Sequential-modular simulators calculate unit by unit, iterating around recycle loops, while equation-oriented simulators solve all equations simultaneously, which handles complex recycles and optimisation more efficiently.
Several kinds of tools are used in the dairy industry. General chemical-process flowsheet simulators offer extensive thermodynamic libraries and unit operation models but were designed for petrochemical systems; batch-process simulators, built for pharmaceutical and food processing, handle batch scheduling, equipment occupancy, cleaning and resource use; specialised food and dairy simulators include models of pasteurisers, evaporators, dryers and membranes with dairy properties; and general numerical environments and programming languages are used to build custom models. Discrete-event simulation tools model plant logistics: tanker arrivals, silo occupancy, production schedules, cleaning and packaging lines.
A recurring challenge is property data. Dairy streams are mixtures of water, lactose, proteins, fat, minerals and other components, and their densities, specific heats, viscosities, boiling point elevations, solubilities and phase behaviours are not available in standard chemical databases. Users define pseudo-components, such as lactose, casein, whey protein, fat and ash, and enter correlations for properties as functions of composition and temperature, drawn from the dairy literature, which must be validated against measurements.
Simulation supports many engineering tasks. In design, it calculates mass and energy balances for new plants and products, sizes equipment such as evaporator effects, heat exchangers and dryers, and estimates steam, electricity, water and refrigeration requirements and capital and operating costs. In debottlenecking, it identifies which unit limits throughput and evaluates options for expanding capacity. In energy studies, it supports heat-recovery design, evaluation of vapour recompression and heat pumps, and pinch analysis. In scheduling, batch and discrete-event simulators optimise production sequences to minimise changeovers, cleaning and idle time, accounting for milk supply variations and cleaning requirements.
Examples include simulating a whey processing plant that produces whey protein concentrate by ultrafiltration and diafiltration, and lactose and permeate powder by evaporation, crystallisation and drying, to compare process configurations; simulating an evaporator with mechanical vapour recompression to determine compressor duty and condensate quality; and simulating a milk powder plant's dryers and fluid beds to evaluate the effect of feed solids and air humidity on capacity.
Dynamic simulation supports control system design and operator training: models of pasteurisers with their control loops can test responses to flow and temperature disturbances, the action of the flow-diversion valve and the effectiveness of control strategies before commissioning, reducing risk. Simulation results are only as good as the models and data; validation against plant measurements, sensitivity analysis and clear documentation of assumptions are essential before decisions rest on them.
A digital twin is a model of a physical asset or process that is continuously updated with data from the real system and used to monitor, predict and optimise its operation. In dairy plants, digital twins of evaporators, dryers, membrane plants and utilities combine first-principles models with plant data streams, using data reconciliation to correct measurement errors and parameter estimation to track changing conditions such as fouling. They support predictive maintenance, energy optimisation, soft sensing of unmeasured variables, and what-if analysis before changes are made to the real plant.
Building and maintaining models requires discipline. Models must be documented, with their assumptions, equations, parameter sources and validation ranges; changes must be version-controlled; and responsibility for keeping models consistent with the real plant, as equipment is modified, must be assigned. Hybrid models, combining mechanistic equations with data-driven components that learn residual behaviour, are increasingly used where physics is incompletely known.
Software selection considers capabilities for dairy properties and unit operations, batch and continuous modes, dynamic simulation, integration with plant data systems, cost, user skills and vendor support. Above all, engineers using simulation must understand the underlying physics well enough to recognise when results are wrong, because a converged simulation can still be meaningless if inputs or models are inappropriate.
Pressure vessels, such as UHT holding vessels, evaporator bodies, steam and condensate vessels, autoclaves, compressed-air receivers and refrigeration receivers, store large amounts of energy and must be designed, manufactured, inspected and operated to recognised codes. Major codes include the American Society of Mechanical Engineers Boiler and Pressure Vessel Code, whose Section VIII covers unfired pressure vessels; the European standard EN 13445 for unfired pressure vessels, used with the European Pressure Equipment Directive; and national codes in many other countries, often derived from or recognised as equivalent to these. Shell-and-tube heat exchangers are additionally designed to standards such as those of the Tubular Exchanger Manufacturers Association, and pressure equipment regulations classify equipment into categories by pressure, volume and fluid hazard, determining the required conformity assessment.
Codes specify design pressure and temperature, which must cover the most severe coincident conditions in service, including vacuum; allowable stresses for each material and temperature, derived from tensile and yield strength with safety factors; weld joint efficiencies, which reduce allowable stress according to the type and extent of weld inspection; corrosion allowances added to calculated thicknesses; rules for openings, nozzles, flanges and supports; material requirements and certification; fabrication, welding and inspection requirements; and pressure testing.
A designer must use the code that applies in the jurisdiction of installation, together with any additional requirements of the purchaser, insurer or regulator; equipment designed to one code is not automatically acceptable elsewhere.
For a cylindrical shell under internal pressure, membrane theory gives a hoop stress of PD/(2t), twice the longitudinal stress. Codes refine this into design formulas. In the widely used formula of ASME Section VIII Division 1 for circumferential stress in thin cylindrical shells, the required thickness is t = PR/(SE − 0.6P), where P is the design pressure, R the inside radius, S the allowable stress and E the joint efficiency. For example, a stainless-steel vessel with an inside radius of 1,000 mm, a design pressure of 0.3 MPa, an allowable stress of 138 MPa and a joint efficiency of 0.85 needs t = 0.3 × 1,000/(138 × 0.85 − 0.18) ≈ 2.6 mm, before adding any corrosion allowance and before considering minimum thicknesses for fabrication, handling and other loads. Heads have their own formulas; a hemispherical head needs about half the thickness of a cylinder of the same radius, and a 2:1 semi-ellipsoidal head roughly the same as the cylinder.
Vessels under vacuum or external pressure, such as evaporator bodies and vacuum vessels, fail by buckling rather than by yielding, and codes provide charts relating allowable external pressure to the vessel's length, diameter and thickness; stiffening rings are added to shorten the effective length. Dairy vacuum vessels, often thin-walled stainless steel for hygiene, need careful buckling design.
Before service, vessels are pressure-tested: hydrostatic testing, with water at a pressure above the design pressure, commonly about 1.3 times the maximum allowable working pressure under current editions of the ASME code and with different factors in other codes, is preferred because water stores little energy if a failure occurs; pneumatic tests at lower multiples are used only when hydrostatic testing is impractical, with additional precautions. Every pressure vessel must be protected by pressure-relief devices sized for the worst credible overpressure scenario, such as fire, blocked outlet or control failure.
Heat exchanger design combines thermal and hydraulic sizing with mechanical design. Thermal design calculates the area required from the duty, overall coefficient and mean temperature difference, including fouling allowances; hydraulic design ensures that pressure drops are acceptable and velocities sufficient for heat transfer and to limit fouling. For plate heat exchangers, design uses manufacturer correlations for each plate pattern, and the maximum pressure and temperature are limited by the frame, plates and gaskets, typically to around 1 to 2.5 MPa and to temperatures compatible with the gasket elastomer; higher-pressure and higher-temperature duties use tubular or welded designs.
Shell-and-tube exchangers are classified by front head, shell and rear head types in standards, and their mechanical design covers tube-sheet thickness, tube vibration, thermal expansion and differential pressure. Double-wall or double-tubesheet designs are used where any leakage between fluids must be detected or prevented, for example where a heat-transfer fluid must never contaminate the product.
For dairy equipment, hygienic design requirements apply in addition to pressure codes. Guidelines from organisations such as the European Hygienic Engineering and Design Group and 3-A Sanitary Standards specify materials, surface finishes, drainability, absence of crevices, cleanability and inspection. Regulatory requirements for pasteurisers, such as pressure differentials between pasteurised and raw milk in regenerators, leak testing of plates, and requirements for flow-diversion and recording equipment, also shape heat exchanger design. The engineer must satisfy pressure safety, thermal performance, hygiene and regulatory requirements simultaneously, which often involves trade-offs, such as between thin plates for heat transfer and robustness for pressure and inspection.
Most dairy process control uses proportional–integral–derivative (PID) loops, tuned by methods such as Ziegler–Nichols, Cohen–Coon or internal model control rules, or by auto-tuning. Several structures improve on single loops. Cascade control nests a fast inner loop inside a slower outer loop: in a pasteuriser, the outer loop controls the milk outlet temperature by setting the setpoint of an inner loop that controls the hot-water temperature by steam, so disturbances in steam pressure are corrected before they affect the milk. Feedforward control measures a disturbance, such as a change in milk flow or inlet temperature, and adjusts the manipulated variable in anticipation, before the controlled variable deviates. Ratio control keeps two flows in a set proportion, as in in-line standardisation, where cream is remixed with skim milk in a ratio controlled from in-line fat measurement. Override and split-range control handle constraints and multiple actuators.
Model predictive control (MPC) uses a dynamic model of the process to predict future behaviour over a horizon and calculates the sequence of control moves that optimises a cost function, such as tracking setpoints with minimum energy, while respecting constraints on inputs and outputs; only the first move is applied, and the calculation repeats at each step. MPC handles multivariable interactions and constraints naturally, which suits evaporators and spray dryers, where feed rate, steam pressure, inlet air temperature and outlet temperature interact, and the goal is to maximise throughput while keeping product moisture and solids within specification. MPC has been applied successfully in dairy evaporators and dryers, often giving tighter control that allows operation closer to constraints and higher capacity.
Many important quality variables, such as powder moisture, concentrate solids, protein content or degree of fouling, are measured infrequently in the laboratory or not at all online. Soft sensors, or inferential sensors, estimate them continuously from routinely measured variables using models: mechanistic models, such as dryer heat and mass balances that estimate powder moisture from exhaust humidity and temperatures, or data-driven models, such as partial least squares regression on many process variables. Soft sensors provide feedback for control between laboratory analyses and must be recalibrated when they drift.
Multivariate statistical process control extends control charts to many correlated variables. Principal component analysis of normal operating data defines a model of normal correlation structure; new data are projected onto it, and two statistics are monitored: Hotelling's T², which detects unusual variation within the model space, and the squared prediction error, which detects new kinds of variation that break the normal correlations. Contribution plots identify which variables cause an alarm. Such monitoring detects faults, such as sensor failures, fouling, or raw-material changes, earlier than univariate charts, and has been applied to pasteurisers, evaporators, dryers and fermentations. Batch processes, such as cheese vats and fermentations, are monitored with multiway methods that compare each batch trajectory with good past batches.
In-line analysers, such as near-infrared spectroscopy for solids, fat and protein, provide the measurements that make these methods effective, forming part of what is often called process analytical technology.
Machine learning methods learn relationships from data rather than from physical models. In dairy plants, applications include predicting product quality from process data, for example cheese moisture or powder solubility; predicting equipment failures from vibration, temperature and current signals for predictive maintenance; detecting anomalies in sensor data; forecasting milk intake and demand for planning; optimising cleaning schedules from fouling indicators; and image analysis for packaging and foreign-body inspection. Methods range from regression trees, random forests and gradient boosting to neural networks, with reinforcement learning studied for control.
Successful applications share features: large, well-labelled, representative data; clear definition of the problem and the value of better predictions; validation on data from periods and conditions not used in training; and integration into decisions, with operators understanding and trusting the outputs. Common failures arise from training on data that do not represent future conditions, such as a new season or modified equipment; from data leakage, where information about the outcome inadvertently enters the inputs; from lack of interpretability; and from neglect of model maintenance as processes drift.
Safety-critical functions must not depend on machine learning or on optimisation software that could fail unpredictably. Food safety controls, such as pasteurisation temperature limits, flow diversion and holding time, are implemented in independent, validated control and safety systems with simple, verifiable logic; process safety functions, such as protection against over-pressure or ammonia release, follow standards for safety instrumented systems, such as IEC 61511. Advanced control and machine learning operate within these boundaries, optimising performance but never overriding protective layers.
Pinch analysis, developed by Linnhoff and colleagues from the late 1970s, is a systematic method for maximising heat recovery between process streams and minimising external heating and cooling. Each stream that must be heated (a cold stream) or cooled (a hot stream) is described by its supply and target temperatures and its heat capacity flow rate, CP = ṁc_p. Hot streams are combined into a hot composite curve and cold streams into a cold composite curve on a temperature–enthalpy diagram. The curves are moved horizontally until, at their closest vertical approach, they are separated by the minimum temperature difference allowed for heat exchange, ΔT_min. The overlap of the curves is the maximum possible heat recovery; the parts that do not overlap are the minimum hot utility, such as steam, and minimum cold utility, such as chilled water or refrigeration.
The point of closest approach is the pinch. It divides the process into a region above the pinch, which needs net heat input, and a region below, which rejects net heat. Three rules follow for an optimal network: do not transfer heat across the pinch, do not use cold utility above the pinch, and do not use hot utility below the pinch. Violating any rule increases both hot and cold utility by the same amount.
The targets are calculated before design by the problem table algorithm. Stream temperatures are shifted by ΔT_min/2, hot down and cold up, temperature intervals are formed, the heat surplus or deficit in each interval is calculated, and heat is cascaded down from the highest interval. The largest deficit found in the cascade is the minimum hot utility, and the heat remaining at the bottom is the minimum cold utility.
Dairy plants have many heating and cooling duties at moderate temperatures: pasteurisation with regeneration, CIP solution heating, evaporator preheating and condensing, dryer inlet air heating, hot water for cleaning and utilities, and cooling of milk, products and refrigeration condensers. Pinch analysis of whole sites typically shows that heat recovery within individual units, such as pasteuriser regeneration, is good, but that recovery between units, such as using evaporator condensate or refrigeration condenser heat to heat CIP water or boiler feed water, is incomplete.
The grand composite curve, which plots the net heat flow against shifted temperature, shows where and at what temperature utilities are needed and guides the selection of utilities and the placement of heat pumps. A heat pump saves energy only if it takes heat from below the pinch and delivers it above; placed entirely on one side, it merely converts electricity into heat. In dairies, where the pinch is often at modest temperatures, heat pumps that upgrade low-temperature waste heat, from refrigeration condensers, exhaust air or cooling water, to supply hot water at 60 to 90 °C for cleaning and processing are often well placed, and they are central to electrification strategies that replace fossil-fuel boilers. Mechanical vapour recompression in evaporators is a special case of heat pumping across the evaporator's own small temperature difference.
Choosing ΔT_min involves an economic trade-off: a smaller ΔT_min increases heat recovery and reduces utility costs but requires more heat-exchanger area; the optimum is found by plotting total annualised cost against ΔT_min. In dairy applications with liquid–liquid exchangers, optimal values are often a few kelvin to around ten kelvin.
Water pinch analysis applies similar ideas to water use. Each water-using operation is characterised by its limiting inlet and outlet contaminant concentrations, such as the maximum chemical oxygen demand or conductivity of water it can accept and produce, and its contaminant load. Plotting these as a limiting composite curve against a water supply line identifies the minimum freshwater flow and the opportunities for reuse, in which water from one operation is used directly in another that tolerates its quality; regeneration, in which water is treated and then reused; and recycling. Dairies have a particular source of water in the vapour condensate from evaporating milk, sometimes called cow water, which after polishing by reverse osmosis and disinfection can replace a large share of freshwater use for cleaning, boiler feed and cooling, and in some plants allows near water self-sufficiency.
Sustainability engineering uses metrics and frameworks to set targets and measure progress. Plant indicators include energy use per tonne of milk or product, water use per litre of milk, effluent load, and greenhouse gas emissions divided into scope 1 (direct emissions from fuel combustion and refrigerant leaks), scope 2 (purchased electricity and heat) and scope 3 (supply chain emissions, dominated in dairy by on-farm emissions from enteric methane, manure and feed). Life-cycle assessment, standardised in ISO 14040 and 14044, quantifies environmental impacts from farm to disposal and shows that processing typically accounts for a minority of dairy product footprints, with farm emissions dominating, although energy-intensive products such as powders have larger processing shares.
Engineers contribute by heat integration, electrification with heat pumps and renewable electricity, efficient refrigeration with low-global-warming-potential refrigerants such as ammonia and carbon dioxide, water reuse, effluent treatment with biogas recovery, and design of new plants for low energy and water use from the start.
Build the models yourself: integrate a real pasteuriser or UHT temperature log to obtain lethality and B* and C*; fit a resistance-in-series and film model to ultrafiltration flux data; calculate freezing time with Plank's equation and compare it with a measurement; set up a small flowsheet with user-defined dairy components; and work a problem-table pinch analysis for four streams. Doing these by hand first, then in software, builds the judgement to recognise wrong outputs before anyone acts on them in a real plant.
A process engineer five years into the job validates heat treatments from data loggers, investigates fouling and optimises cleaning cycles, sizes membrane plants and diafiltration schedules, tunes cascade and feedforward loops on pasteurisers and evaporators, specifies pressure vessels and heat exchangers to the applicable codes, and uses simulation to evaluate capacity and energy projects. Typical decisions: whether run length can be extended without risk, how many diavolumes are needed for a target lactose level, and whether a proposed heat pump crosses the pinch.
An engineering director with 25 years' experience sets the technical strategy for plant investments: evaporator and dryer configurations, electrification and heat pump programmes, water reuse, advanced control and digital twin projects, and compliance with pressure equipment and hygiene requirements across sites in different jurisdictions. They judge models and vendor claims critically, prioritise projects by return and risk, and ensure that optimisation never compromises food safety or process safety systems, and that knowledge is retained when experienced engineers retire or move on.
A professor or research engineer advances modelling of dairy processes: mechanistic fouling and cleaning models, particle formation in spray drying with CFD and glass-transition criteria, membrane fouling and polarisation in complex dairy streams, stochastic lethality models, freezing and recrystallisation kinetics, and hybrid physics–data models for control. They publish validated models with data, collaborate with industry on scale-up, and teach the next generation to combine rigorous transport phenomena with practical engineering judgement, and serve on committees that write the codes and guidelines.
Design formulas from codes are quoted only as illustrative examples; actual design must follow the current edition of the code applicable in the jurisdiction, with all its provisions. Numerical values (D, z, B* and C* definitions, fouling temperature ranges, membrane operating conditions) are standard literature values that vary with product and source. Detailed CFD methodology, numerical analysis, control theory proofs and machine learning algorithms are introduced but not developed. Specific commercial software is not named or taught; capabilities change frequently.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions.
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