Dairy Science & Technology · Dairy Engineering & Plant Operations · also Dairy Engineering I, Fluid Mechanics, Heat and Mass Transfer
Fluid properties, pumping, thermodynamics and heat transfer worked as dairy plant problems.
In the exams: ICAR AICE subject 38 Units 1 and 4; ICAR AIEEA-PG group 17 Unit III (fluid mechanics, pumps, heat transfer, thermodynamics, dimensional analysis); GATE XL-U Section 4; GATE AG dairy and food engineering
Every dairy process moves fluids and moves heat. Milk is pumped from tanker to silo, through separators, pasteurisers and homogenisers, into fillers; cream, yoghurt and concentrates, which flow very differently from milk, must be pumped without damage; steam, hot water, chilled water and refrigerants carry heat to and from the product; and every heating, cooling, evaporating and drying step must be designed with the laws of thermodynamics, heat transfer and mass transfer. This subject applies the general theory of fluid mechanics, thermodynamics and transport phenomena, taught fully in the agricultural engineering track, to dairy plant problems.
The manual covers eleven topics: fluid properties and Bernoulli's equation; pipe flow, the Reynolds number and friction losses; sanitary centrifugal and positive-displacement pumps and how to select them; the flow of non-Newtonian dairy fluids; the laws of thermodynamics and the use of steam tables; conduction, convection and radiation; fins and insulation; heat exchangers analysed by the log-mean temperature difference and the effectiveness–NTU methods; diffusion and mass transfer coefficients; dimensionless numbers and dimensional analysis; and mass and energy balances of dairy processes. Each is worked with dairy numbers: milk at about 1,030 kg/m³, a viscosity near 2 mPa·s and a specific heat near 3.9 kJ/kg K.
Postgraduate entrance and doctoral fellowship examinations in dairy engineering, and graduate aptitude tests in food technology and agricultural engineering, set numerical problems on exactly these topics: Bernoulli and continuity, Reynolds number and head loss, pump power, power-law flow, steam quantities from steam tables, conduction through walls and pipes, overall heat-transfer coefficients, LMTD and regeneration efficiency, Fick's law and mass and energy balances. The worked examples here show the method in full.
A fluid is a substance that deforms continuously under shear stress. The properties that govern its flow are density, viscosity and, for gases, compressibility. Density ρ is mass per unit volume; water at 20 °C is about 998 kg/m³, whole milk about 1,028 to 1,034 kg/m³ and cream lower, falling as fat content rises, because milk fat is lighter than the serum.
Viscosity measures resistance to flow. In a Newtonian fluid, shear stress τ is proportional to shear rate: τ = μ(du/dy), where μ is the dynamic viscosity in pascal-seconds. Water at 20 °C has μ ≈ 1.0 mPa·s; whole milk about 2 mPa·s, rising to about 3 mPa·s at 4 °C and falling to under 1 mPa·s near pasteurisation temperatures, because viscosity of liquids falls steeply with temperature. Kinematic viscosity ν = μ/ρ, in m²/s, appears in flow calculations. Milk and skim milk behave as Newtonian fluids at ordinary concentrations; cream, concentrates, yoghurt and many desserts do not, as described in a later chapter.
Pressure is force per unit area, in pascals; 1 bar = 100 kPa. Gauge pressure is measured relative to the atmosphere and absolute pressure relative to vacuum; absolute = gauge + atmospheric, about 101.3 kPa at sea level. Hydrostatic pressure in a liquid at depth h is p = ρgh, so a silo of milk 10 m deep exerts about 1,030 × 9.81 × 10 ≈ 101 kPa at its base, roughly one atmosphere above the surface pressure.
Conservation of mass in steady flow through a pipe gives the continuity equation: the mass flow rate ṁ = ρAv is constant along the pipe, so for an incompressible liquid the volumetric flow rate Q = Av is constant and velocity rises where the pipe narrows. Milk flowing at 10,000 L/h (2.78 × 10⁻³ m³/s) in a pipe of 51 mm internal diameter (area 2.04 × 10⁻³ m²) moves at about 1.36 m/s; in a 38 mm pipe the same flow moves at about 2.45 m/s.
Conservation of energy for an ideal, frictionless, incompressible fluid along a streamline gives Bernoulli's equation: p/ρg + v²/2g + z = constant, where the three terms are the pressure head, velocity head and elevation head, each in metres of the fluid. For real flow between two points, an energy equation adds the head supplied by a pump, H_p, and the head lost to friction, h_L: p₁/ρg + v₁²/2g + z₁ + H_p = p₂/ρg + v₂²/2g + z₂ + h_L.
Bernoulli's principle underlies flow measurement. In a Venturi meter or orifice plate, the fluid speeds up in the constriction and its pressure falls; the pressure difference gives the velocity, v₂ = √(2Δp/ρ(1 − (A₂/A₁)²)), multiplied by a discharge coefficient for real losses. Flow out of a tank through an outlet at depth h below the free surface has the Torricelli velocity √(2gh), which governs gravity draining of vats.
Bernoulli and continuity explain many plant observations. A pipe that narrows at a valve or fitting has higher velocity and lower pressure there; if the pressure falls to the vapour pressure of the liquid, bubbles form and collapse, which is cavitation, a cause of noise, damage and air incorporation, especially in hot milk near pump inlets. Suction lines should therefore be short, wide and free of restrictions.
Pressure differences are deliberately managed in pasteurisers. In a plate heat exchanger with regeneration, the pasteurised milk must be at a higher pressure than the raw milk on the other side of each plate, so that a pinhole leak would let pasteurised milk flow into raw milk rather than the reverse. A booster pump after the regenerator raw side, or placement of the pump and flow restrictions, establishes this difference, and pressure gauges or differential pressure switches confirm it. Liquid levels in tanks are measured by hydrostatic pressure transmitters using p = ρgh, which must be corrected for product density: a transmitter calibrated for water reads about 3% high in milk-level terms if not corrected.
Flow in pipes is either laminar, in which fluid moves in smooth layers with no mixing between them, or turbulent, in which eddies mix the fluid vigorously. Osborne Reynolds showed that the type of flow depends on the dimensionless Reynolds number, Re = ρvD/μ = vD/ν, the ratio of inertial to viscous forces. In pipes, flow is laminar below about Re = 2,100 (some texts use 2,300), turbulent above about 4,000, and transitional between.
Milk at 20 °C (ρ = 1,030 kg/m³, μ = 0.002 Pa·s) flowing at 1.5 m/s in a 50 mm pipe has Re = 1,030 × 1.5 × 0.05/0.002 ≈ 38,600, well into the turbulent range. Most milk flows in dairy plants are turbulent; viscous products such as cream with high fat, concentrates, yoghurt and desserts often flow in the laminar or transitional range.
The velocity profile differs between the two regimes. In laminar flow of a Newtonian fluid the profile is parabolic, with the maximum velocity at the centre equal to twice the mean. In turbulent flow the profile is much flatter, with the maximum about 1.2 times the mean, steep gradients near the wall in a thin laminar sublayer, and nearly uniform velocity across the core. This matters for holding tubes: the fastest particle travels faster than the average, so the minimum residence time is shorter than the mean, and regulations and design practice require the holding tube to be long enough for the fastest particle, assuming the laminar ratio of 2 unless turbulent flow is demonstrated. It matters also for cleaning, because turbulent flow at the wall, usually at least 1.5 m/s in pipelines, is needed to remove deposits.
Friction between the fluid and the pipe wall dissipates energy, appearing as a loss of pressure along the pipe. The Darcy–Weisbach equation gives the head loss: h_f = f(L/D)(v²/2g), where f is the Darcy friction factor, L the pipe length and D its internal diameter; the pressure drop is Δp = ρgh_f.
For laminar flow, the friction factor depends only on Reynolds number: f = 64/Re (Darcy), or 16/Re (Fanning), which is equivalent to the Hagen–Poiseuille law, Δp = 32μLv/D². For turbulent flow, f depends on Reynolds number and the relative roughness of the wall, ε/D, as shown on the Moody chart or calculated from the Colebrook equation. Sanitary stainless-steel tubing is hydraulically smooth, so the Blasius equation, f = 0.316 Re^(−0.25), is a good approximation for Re from about 4,000 to 100,000.
A worked example: milk at Re = 38,600 in a 50 mm pipe at 1.5 m/s over 40 m. Blasius gives f = 0.316 × 38,600^(−0.25) ≈ 0.0225. Then h_f = 0.0225 × (40/0.05) × (1.5²/(2 × 9.81)) ≈ 0.0225 × 800 × 0.115 ≈ 2.06 m of milk, a pressure drop of about 1,030 × 9.81 × 2.07 ≈ 20.9 kPa. Because h_f varies roughly with v^1.75 in turbulent smooth-pipe flow, and with v in laminar flow, doubling velocity in turbulent flow raises friction loss more than threefold.
Fittings, valves, bends, tees, reducers, entrances and exits cause additional losses, called minor losses although in short dairy pipelines with many fittings they can exceed pipe friction. They are expressed as h_m = K(v²/2g), where K is a loss coefficient from tables for each fitting, or as an equivalent length of straight pipe. Typical K values are about 0.5 for a sharp pipe entrance, 1.0 for an exit into a tank, 0.3 to 0.9 for standard bends depending on radius, 0.2 for an open gate-type valve, several for globe-type valves, and significant values for flow meters, strainers and plate heat exchangers, whose pressure drop is usually given by the manufacturer for the flow rate.
The total head a pump must supply is the system head: the static head (the difference in elevation and pressure between the source and destination) plus all friction and minor losses, which rise with the square of flow for turbulent flow. Plotted against flow, this gives the system curve, a parabola starting at the static head. The pump operates where its own head–flow curve crosses the system curve.
Pipe sizing is a balance. Larger pipes have lower velocities and losses, saving pumping energy, but cost more, hold more product at changeover and may not reach the velocity needed for effective cleaning. Dairy lines are typically sized for product velocities of about 1 to 2 m/s, with cleaning flows of at least 1.5 m/s, and with care to avoid dead legs and to ensure the line drains.
The centrifugal pump is the commonest pump in dairies. An impeller spinning inside a casing throws liquid outwards, converting the energy of rotation into velocity and then, in the volute, into pressure. Sanitary centrifugal pumps have stainless-steel open or semi-open impellers, smooth casings that can be cleaned in place, hygienic shaft seals, usually single or double mechanical seals, and drainable designs. They suit low-viscosity liquids such as milk, whey, cream of moderate fat, water and cleaning solutions, and they are simple, cheap and tolerant of a closed outlet for short periods.
A centrifugal pump's performance is shown by its characteristic curves: head against flow, which falls as flow rises; efficiency, which peaks at the best efficiency point; and power. The pump delivers the flow at which its head curve meets the system curve; throttling a valve on the outlet raises the system curve and reduces flow. The hydraulic power is ρgQH, and the shaft power P = ρgQH/η, where η is the pump efficiency. A pump delivering 20 m³/h of milk (5.56 × 10⁻³ m³/s) against 25 m of head at 60% efficiency needs 1,030 × 9.81 × 5.56 × 10⁻³ × 25/0.6 ≈ 2.34 kW.
The affinity laws relate performance to speed for a given impeller: flow is proportional to speed, head to speed squared and power to speed cubed. They explain the energy savings of variable-speed drives. Centrifugal pumps are not self-priming unless specially designed; they must be flooded with liquid at start-up, so they are placed below the source tank.
Positive displacement pumps trap a fixed volume of liquid and force it to the outlet with each revolution or stroke, so their flow depends on speed and is nearly independent of the discharge pressure. They are used for viscous products, for gentle handling, for accurate metering and for very high pressures. Because they keep delivering against a closed outlet until something breaks, they must be fitted with pressure relief or bypass arrangements.
Rotary lobe pumps, with two rotating lobed rotors that do not touch, are widely used in dairies for cream, yoghurt, quarg, curd, concentrates and ice cream mix, because they handle viscous and particulate products gently, are cleanable in place and can run in either direction. Progressing cavity pumps, with a helical rotor in an elastomer stator, move very viscous or shear-sensitive products smoothly. Reciprocating piston and plunger pumps produce the very high pressures, typically up to 25 MPa or more, needed for homogenisers.
In HTST pasteurisers, the flow rate through the holding tube must not exceed the rate on which the holding time was calculated. Traditionally a positive-displacement timing pump fixed the maximum flow; modern systems often use a centrifugal pump with a flow meter and control valve, with flow-alarm and diversion logic, accepted by regulators where it is shown to be at least as reliable.
Selecting a pump starts with the duty: the product and its properties at the pumping temperature (viscosity, density, solids, shear sensitivity), the flow rate, the total head from the system curve, the suction conditions, the hygienic requirements and the cleaning regime. Low-viscosity liquids at moderate heads suit centrifugal pumps; viscous products, high pressures, metering and shear-sensitive products suit positive-displacement pumps. Centrifugal pump performance falls sharply as viscosity rises, and correction charts are used above about 50 to 100 mPa·s.
Suction conditions are critical. If the pressure at the pump inlet falls below the vapour pressure of the liquid, vapour bubbles form and then collapse violently as pressure rises inside the pump: cavitation. It causes noise, vibration, loss of performance and pitting of impellers, and in dairy pumps it also damages product by incorporating air and denaturing proteins. To avoid it, the net positive suction head available, NPSH_A, must exceed the net positive suction head required by the pump, NPSH_R, given by the manufacturer, with a margin. NPSH_A = (p_surface − p_vapour)/ρg + z_s − h_f,s, where z_s is the height of the liquid surface above the pump inlet (negative if below) and h_f,s the friction loss in the suction line.
Hot liquids are the main risk, because vapour pressure rises steeply with temperature: water at 20 °C has a vapour pressure of about 2.3 kPa, at 80 °C about 47 kPa and at 100 °C atmospheric pressure. Pumping hot milk or cleaning solutions from an open tank needs a short, wide suction line and a good height of liquid above the pump.
Many dairy fluids are non-Newtonian: their apparent viscosity, the ratio of shear stress to shear rate, depends on the shear rate, the time of shearing, or both. Shear-thinning (pseudoplastic) fluids become less viscous as shear rate rises; examples are concentrated milk, cream of high fat, yoghurt, ice cream mix, many desserts and solutions of hydrocolloid stabilisers. Some fluids have a yield stress, a minimum stress below which they do not flow, like set yoghurt, cream cheese and some concentrates.
The power-law (Ostwald–de Waele) model describes many shear-thinning fluids: τ = Kγ̇ⁿ, where K is the consistency coefficient (Pa·sⁿ) and n the flow behaviour index; n < 1 is shear-thinning, n = 1 Newtonian and n > 1 shear-thickening. The apparent viscosity is μ_app = Kγ̇^(n−1). A yoghurt with K = 10 Pa·sⁿ and n = 0.3 has an apparent viscosity of 10 × 10^(−0.7) ≈ 2.0 Pa·s at a shear rate of 10 s⁻¹ but only about 0.40 Pa·s at 100 s⁻¹. The Bingham plastic model, τ = τ₀ + μ_pγ̇, and the Herschel–Bulkley model, τ = τ₀ + Kγ̇ⁿ, include a yield stress τ₀.
Time-dependent fluids change with shearing time at constant shear rate. Thixotropic fluids, such as stirred yoghurt, lose viscosity when sheared and recover partly or slowly at rest, because the protein network is broken and only partly rebuilds; rheopectic fluids, rare in foods, do the opposite.
For laminar flow of a power-law fluid in a pipe, the velocity profile is flatter than the parabola of a Newtonian fluid, and the ratio of maximum to mean velocity is u_max/ū = (3n + 1)/(n + 1). For n = 1 this gives 2, the Newtonian value; for a yoghurt with n = 0.3 it gives 1.9/1.3 ≈ 1.46. This flatter profile matters for holding tubes and for the residence time distribution of particles in viscous products.
The Reynolds number for power-law fluids is defined by Metzner and Reed: Re_MR = ρv^(2−n)Dⁿ/(K 8^(n−1) ((3n + 1)/4n)ⁿ), which reduces to the ordinary Reynolds number when n = 1. With this definition, laminar flow has a Fanning friction factor of 16/Re_MR, as for Newtonian fluids, and the transition to turbulence occurs at a value that depends on n but is often taken as about 2,100. Most viscous dairy products flow in the laminar regime in plant pipes because their apparent viscosities are hundreds to thousands of times that of milk.
Shear-thinning helps pumping: the high shear rates near the pipe wall lower the apparent viscosity there, reducing the pressure drop compared with a Newtonian fluid of the same low-shear viscosity. Yield stress fluids may flow as a plug in the centre of the pipe, with shearing only near the wall. Pressure drop calculations for such products use the appropriate model and rheological data measured at the processing temperature.
Rheological properties are measured with viscometers and rheometers. Capillary and tube viscometers measure flow rate and pressure drop in a tube and calculate the wall shear stress and shear rate, with a correction for non-Newtonian behaviour. Rotational viscometers measure torque on a spindle, cylinder, cone or plate rotating in the sample at controlled speed; concentric-cylinder and cone-and-plate geometries give well-defined shear rates.
Rheology governs many dairy process decisions. The choice between centrifugal and positive-displacement pumps depends on viscosity; heat-exchanger type depends on flow behaviour, with plate heat exchangers suited to low-viscosity products and tubular or scraped-surface heat exchangers to viscous or particulate ones; holding-tube length depends on the velocity profile; and pipeline pressure drops determine pump heads. Shear in pumps, valves and fillers breaks the structure of stirred yoghurt, so its viscosity after filling may be far below that in the fermentation tank; low-shear pumps, gentle valves, wide pipes and smooth coolers are used to preserve texture.
Temperature changes rheology strongly. Milk concentrates thicken as total solids rise in the last effect of an evaporator, and age-thicken during storage; cream thickens markedly when cooled as fat crystallises; and ice cream mix viscosity rises on ageing as stabilisers hydrate and fat crystallises. Rheological data must therefore be obtained at the actual process conditions.
Thermodynamics deals with energy and its transformations. The zeroth law states that two bodies each in thermal equilibrium with a third are in equilibrium with each other; it is the basis of temperature measurement. The first law is the conservation of energy: for a closed system, the change in internal energy equals heat added minus work done by the system, ΔU = Q − W. For an open, steady-flow system such as a heat exchanger, pump or evaporator, the first law becomes an energy balance on enthalpy: Q − W = ṁ(Δh + Δv²/2 + gΔz), and in most dairy heating and cooling problems, kinetic and potential energy changes are negligible, so Q = ṁΔh. For a liquid without phase change, Δh = c_p ΔT; milk's specific heat is about 3.9 kJ/kg K and water's 4.18 kJ/kg K.
The second law states that heat flows spontaneously only from hot to cold and that no cyclic engine can convert heat entirely into work. It introduces entropy, which in any real process increases for the system and its surroundings together. The maximum efficiency of a heat engine operating between absolute temperatures T_H and T_C is the Carnot efficiency, 1 − T_C/T_H, and the maximum coefficient of performance of a refrigerator is T_C/(T_H − T_C). These limits explain why refrigeration at low temperatures costs more energy per unit of heat removed, and why heat recovery and heat pumps are effective ways to save energy.
Steam is the main heating medium in dairies because it carries a large amount of energy per kilogram as latent heat, condenses at a constant temperature set by its pressure, gives very high heat-transfer coefficients on condensation and is easily distributed. When water is heated at constant pressure, its temperature rises until it reaches the saturation temperature for that pressure; further heat converts it to steam at that temperature, absorbing the latent heat of vaporisation; and once all is vapour, further heating produces superheated steam.
Steam tables list, for saturated conditions at each temperature or pressure, the specific enthalpy of the liquid (h_f), the latent heat (h_fg), the enthalpy of the dry saturated vapour (h_g = h_f + h_fg), specific volumes and entropies. Representative values, from standard tables, are: at 101.3 kPa absolute, saturation temperature 100 °C, h_f 419 kJ/kg, h_fg 2,257 kJ/kg; at 200 kPa absolute, 120.2 °C, h_fg about 2,202 kJ/kg; at 300 kPa, 133.5 °C, about 2,164 kJ/kg; at 400 kPa, 143.6 °C, about 2,133 kJ/kg. The latent heat falls as pressure rises, while the saturation temperature rises.
Steam in plants is often wet, containing droplets of water. Its quality or dryness fraction x is the mass fraction of vapour, and its enthalpy is h = h_f + x h_fg. Wet steam carries less latent heat per kilogram and should be dried by separators before use.
Most heating calculations combine the sensible heat of the product with the latent heat of condensing steam. To heat 5,000 kg/h of milk from 60 °C to 90 °C, the heat required is Q = ṁc_pΔT = 5,000 × 3.9 × 30 = 585,000 kJ/h, or 162.5 kW. If saturated steam at 200 kPa absolute condenses and leaves as condensate at saturation temperature, each kilogram gives up h_fg ≈ 2,202 kJ, so the steam needed is 585,000/2,202 ≈ 266 kg/h. If the condensate is subcooled further, a little more heat per kilogram is available; if the steam is wet with x = 0.95, each kilogram gives only 0.95 × 2,202 ≈ 2,092 kJ, and 280 kg/h are needed.
Direct steam heating, by injection or infusion in UHT plants, mixes steam with the product, so the condensate dilutes it. The mass of condensate equals the steam used, and it is removed afterwards by flash cooling in a vacuum vessel, where the hot product boils at the lower pressure and loses vapour; the vacuum is set so that exactly as much water evaporates as was added. An energy balance shows the amount: heating 1 kg of milk by 70 K needs about 3.9 × 70 = 273 kJ, which at an effective 2,100 kJ per kilogram of steam adds about 0.13 kg of water per kilogram of milk.
Steam tables also govern sterilisation and evaporation. An autoclave or retort at 121 °C needs saturated steam at about 205 kPa absolute; any air in the chamber lowers the temperature at a given pressure, which is why air is removed. Evaporators boil milk at reduced pressure to keep its temperature low, and the latent heat at the boiling pressure sets the steam needed for each kilogram of water removed.
Heat is transferred by conduction, convection and radiation. Conduction is the transfer of heat through a material by molecular interaction, without bulk movement. Fourier's law states that the heat flux is proportional to the temperature gradient: q = −k dT/dx, where k is the thermal conductivity in W/m K. Metals conduct well: copper about 400, aluminium about 200, carbon steel about 45 and stainless steel about 15 to 16 W/m K. Water conducts about 0.6, milk slightly less, fats about 0.17, still air about 0.026 and insulation materials about 0.03 to 0.05 W/m K.
For steady conduction through a plane wall of thickness L and area A, Q = kAΔT/L. The thermal resistance is R = L/(kA), so walls of several layers in series add their resistances, like electrical resistors: Q = ΔT/ΣR. For a cylindrical pipe wall of length L between radii r₁ and r₂, Q = 2πkLΔT/ln(r₂/r₁). A worked example: a stainless-steel plate 0.6 mm thick (k = 16 W/m K) with 1 m² area and 1 K across it conducts 16 × 1 × 1/0.0006 ≈ 26,700 W; the resistance of the steel is tiny compared with those of the liquid films on each side, which is why thin stainless plates make efficient heat exchangers despite the metal's modest conductivity.
Unsteady conduction, in which temperature changes with time, governs the heating and cooling of solid or stationary products: cheese blocks in brine, butter in cold stores, milk in containers without stirring. It is described by ∂T/∂t = α∂²T/∂x², where α = k/(ρc_p) is the thermal diffusivity, and solved by charts, series solutions or numerical methods.
Convection is the transfer of heat between a surface and a moving fluid, combining conduction in the fluid near the wall with transport by fluid motion. Newton's law of cooling expresses it as Q = hAΔT, where h is the convective heat-transfer coefficient in W/m² K. In natural or free convection, the motion is caused by density differences from temperature differences, as in an unstirred tank; in forced convection, it is caused by a pump, fan or agitator.
Heat-transfer coefficients vary over a huge range: about 5 to 25 W/m² K for natural convection in air; 10 to 200 for forced air; 100 to 1,000 for natural convection in water; 500 to 10,000 or more for forced convection of water and milk; and 5,000 to 100,000 for condensing steam and boiling water. Coefficients are calculated from correlations between dimensionless numbers, described in a later chapter; for turbulent flow in pipes, the Dittus–Boelter equation Nu = 0.023 Re^0.8 Pr^n, with n = 0.4 for heating and 0.3 for cooling the fluid, is widely used.
In heat exchangers, heat passes from the hot fluid to the wall by convection, through the wall by conduction, and into the cold fluid by convection. The overall heat-transfer coefficient U combines the resistances: for a thin flat wall, 1/U = 1/h_hot + L/k + 1/h_cold + R_f, where R_f is the fouling resistance of deposits. In milk pasteurisers, U values of a few thousand W/m² K are typical when clean, falling as milk deposits build up; in UHT plants and evaporators, fouling is a major factor limiting run time.
Radiation is the transfer of energy by electromagnetic waves, which needs no medium. Every surface above absolute zero emits thermal radiation. A perfect emitter, a black body, emits energy at a rate given by the Stefan–Boltzmann law: q = σT⁴, where σ = 5.67 × 10⁻⁸ W/m² K⁴ and T is absolute temperature. Real surfaces emit a fraction ε, the emissivity, of black-body radiation; ε is about 0.05 to 0.1 for polished metals and 0.8 to 0.95 for most non-metals, paints and oxidised surfaces. The net exchange between a small surface and large surroundings is Q = εσA(T_s⁴ − T_sur⁴).
Because of the fourth power, radiation dominates at high temperatures and is less important at the moderate temperatures of most dairy processes.
A worked example: an uninsulated steam pipe surface at 150 °C (423 K), with emissivity 0.8, in a room at 25 °C (298 K) loses by radiation 0.8 × 5.67 × 10⁻⁸ × (423⁴ − 298⁴) ≈ 0.8 × 5.67 × 10⁻⁸ × (3.20 × 10¹⁰ − 0.79 × 10¹⁰) ≈ 1,090 W per square metre of surface, which shows why insulating steam lines saves so much energy.
Where the heat-transfer coefficient on one side of a wall is much lower than on the other, as with air compared with water or refrigerant, the air side limits the overall rate. Fins, extended surfaces attached to the wall, increase the area on the low-coefficient side. They are used on refrigeration condensers and evaporators, air coolers in cold stores, air heaters for spray dryers, motor housings and electronic components.
A fin does not transfer heat as effectively as if its whole surface were at the base temperature, because its temperature falls along its length as heat flows out through its surface. The fin efficiency η_f is the ratio of the actual heat transfer to the heat that would be transferred if the entire fin were at the base temperature. For a straight fin of uniform cross-section with an insulated tip, η_f = tanh(mL)/(mL), where m = √(hP/kA_c), P is the fin perimeter, A_c its cross-sectional area and L its length. Long, thin fins of low-conductivity material in high h have low efficiency. For example, with mL = 1, η_f = tanh(1)/1 ≈ 0.76; with mL = 2, η_f ≈ 0.48. Fins are therefore made of high-conductivity metals such as aluminium or copper, and kept short.
The heat transfer from a finned surface is Q = h(A_base + η_f A_fin)ΔT. Fin effectiveness, the ratio of heat transfer with the fin to that without it, should be well above one for fins to be worthwhile.
Insulation reduces heat flow by adding a large thermal resistance, using materials of low conductivity, usually those that trap still air or gas in fibres or closed cells: mineral wool, glass wool, polyurethane and polyisocyanurate foam, expanded and extruded polystyrene, and cellular glass, with conductivities of about 0.02 to 0.05 W/m K. Dairy plants insulate steam and hot-water lines, hot tanks and pasteuriser holding tubes against heat loss and burns, and insulate chilled-water lines, ice-bank tanks, silos, cold stores and refrigerant lines against heat gain and condensation.
For a pipe, heat loss through insulation of outer radius r₂ on a pipe of radius r₁ is Q = 2πkL(T_i − T_o)/ln(r₂/r₁), combined in series with the outer convection resistance 1/(h_o 2πr₂L). The thickness is chosen by energy economics, safety (surface temperatures low enough not to burn), and, for cold surfaces, prevention of condensation: the outer surface must stay above the dew point of the surrounding air, or water condenses, wets the insulation and drips onto equipment and products.
Insulation on cold surfaces must have an effective vapour barrier on its warm side. Otherwise water vapour diffuses into the insulation and condenses or freezes there, destroying its insulating value and corroding the pipe beneath. On hot stainless-steel surfaces, insulation materials must be low in chlorides, and cladding must keep out water, because chlorides concentrated under wet insulation cause stress corrosion cracking of austenitic stainless steel.
Adding insulation to a small pipe does not always reduce heat loss. Insulation increases the conduction resistance, but it also increases the outer surface area, which lowers the convection resistance. For a cylinder, heat loss is greatest when the outer radius equals the critical radius, r_c = k/h, where k is the insulation conductivity and h the outer convection coefficient. If the bare pipe radius is smaller than r_c, adding a thin layer of insulation increases heat loss until the outer radius exceeds r_c; beyond that, more insulation reduces it. For typical insulation (k = 0.04 W/m K) and natural convection in air (h = 10 W/m² K), r_c = 0.004 m, or 4 mm, so for almost all process pipes insulation reduces heat loss from the first layer. The critical radius matters mainly for thin electrical wires, where a thin insulating coating can increase heat dissipation.
The economic thickness balances the capital cost of insulation, which rises with thickness, against the cost of energy lost, which falls with thickness. The total annual cost has a minimum at the economic thickness, found by setting the derivative of total cost with respect to thickness to zero or by calculation over standard thicknesses. Higher energy prices, longer operating hours and greater temperature differences justify thicker insulation.
Dairy plants use several types of heat exchanger. Plate heat exchangers, packs of thin corrugated stainless-steel plates sealed by gaskets, give high turbulence and heat-transfer coefficients in a compact unit, are easily opened for inspection and extended, and dominate milk pasteurisation and cooling. Tubular heat exchangers, of concentric tubes or tube bundles, withstand higher pressures and handle more viscous products and particles, and are used in UHT plants.
The rate of heat transfer in a heat exchanger is Q = UAΔT_m, where ΔT_m is a mean temperature difference. For counter-current or co-current flow, the appropriate mean is the log-mean temperature difference: LMTD = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂), where ΔT₁ and ΔT₂ are the temperature differences between the two fluids at the two ends. For example, hot water entering at 90 °C and leaving at 80 °C counter-current to milk heated from 60 °C to 75 °C gives end differences of 90 − 75 = 15 K and 80 − 60 = 20 K, and LMTD = (20 − 15)/ln(20/15) ≈ 17.4 K. If ΔT₁ = ΔT₂, the LMTD equals that difference.
Counter-current flow gives a larger LMTD than co-current flow for the same inlet and outlet temperatures, and it allows the cold fluid to leave hotter than the hot fluid leaves, which co-current flow cannot. Plate heat exchangers are therefore arranged for counter-current flow wherever possible. For multipass and cross-flow arrangements, the LMTD is multiplied by a correction factor F from charts.
In an HTST milk pasteuriser, most of the heat needed to warm the incoming raw milk comes from the hot pasteurised milk leaving the holding tube, in the regenerative section, where the two milks flow counter-current on opposite sides of the plates. The raw milk is heated, and the pasteurised milk is cooled, saving both heating and cooling energy. The regeneration efficiency is the fraction of the total heating supplied by regeneration: η_R = (T_raw,out − T_raw,in)/(T_past − T_raw,in), where T_past is the pasteurisation temperature. For milk entering at 4 °C, leaving the regenerator at 66 °C and pasteurised at 72 °C, η_R = (66 − 4)/(72 − 4) = 62/68 ≈ 91%.
Modern plants achieve regeneration efficiencies of 90 to 95%, so that the heating section supplies only the last few degrees and the cooling section removes only the last few degrees. Higher regeneration needs more plate area and gives a smaller temperature difference, so its gain is balanced against capital cost and the risk that a smaller driving force slows the response to changes. The temperature of the pasteurised milk leaving the regenerator is found from an energy balance: with equal flows and specific heats, the fall in temperature of the pasteurised milk equals the rise of the raw milk, so it leaves at about 72 − 62 = 10 °C, before final cooling to about 4 °C with chilled water.
Regeneration also creates the hygiene requirement that pasteurised milk be at higher pressure than raw milk in the regenerator, so that any leak runs from pasteurised to raw, and the plates must be inspected regularly for pinholes and cracks.
The LMTD method is convenient when all inlet and outlet temperatures are known and the area is sought. When the exchanger and inlet temperatures are known and the outlet temperatures are sought, the effectiveness–NTU method avoids trial and error. The heat-capacity rates are C = ṁc_p for each fluid; C_min is the smaller and C_r = C_min/C_max their ratio. The maximum possible heat transfer is Q_max = C_min(T_h,in − T_c,in), and the effectiveness is ε = Q/Q_max. The number of transfer units is NTU = UA/C_min, a measure of the exchanger's size.
For each flow arrangement, ε is a function of NTU and C_r. For counter-current flow, ε = [1 − exp(−NTU(1 − C_r))]/[1 − C_r exp(−NTU(1 − C_r))], and when C_r = 1, as in a milk-to-milk regenerator with equal flows, ε = NTU/(1 + NTU). For co-current flow, ε = [1 − exp(−NTU(1 + C_r))]/(1 + C_r). When one fluid condenses or boils at constant temperature, C_max is effectively infinite, C_r = 0, and ε = 1 − exp(−NTU) for any arrangement.
A worked example: a regenerator with NTU = 10 and C_r = 1 has ε = 10/11 ≈ 0.91, which corresponds to the 91% regeneration of the previous section; doubling the area to NTU = 20 raises ε only to 0.95, showing diminishing returns. For milk heated by condensing steam with NTU = 2, ε = 1 − e⁻² ≈ 0.86. The method also shows clearly how fouling, which lowers U and hence NTU, reduces the outlet temperature achievable, and why a pasteuriser's hot-water temperature must rise during a run to maintain the pasteurisation temperature.
Mass transfer is the movement of a component from a region of high to low concentration. In molecular diffusion, random molecular motion causes the net movement. Fick's first law gives the flux: J = −D dC/dx, where J is the molar or mass flux per unit area, D the diffusion coefficient or diffusivity in m²/s, and dC/dx the concentration gradient. It is the exact analogue of Fourier's law for heat.
Diffusivities differ by orders of magnitude. In gases they are about 10⁻⁵ m²/s; in liquids about 10⁻⁹ m²/s; and in solids and gels such as cheese, about 10⁻¹⁰ m²/s or less. This explains why salt penetrates cheese so slowly during brining. Fick's second law describes unsteady diffusion, ∂C/∂t = D∂²C/∂x², the analogue of transient heat conduction. A useful estimate from it is that the distance over which concentration changes significantly in time t is of the order of √(Dt): with D = 3 × 10⁻¹⁰ m²/s, in one day (86,400 s) the penetration distance is about √(2.6 × 10⁻⁵) ≈ 5 mm, and in ten days about 16 mm, which is why large cheeses are brined for days and salt continues to equalise for weeks afterwards. Because the time scales with distance squared, halving the size of a cheese reduces brining time to roughly a quarter.
Diffusion through solids and packaging also controls quality: the transfer of oxygen and water vapour through packaging films, the migration of moisture between components of composite products such as cereal inclusions in yoghurt, and the loss of aroma.
When a fluid flows over a surface, mass transfer between them combines diffusion and fluid motion, convective mass transfer. As with convective heat transfer, it is expressed by a coefficient: N = k_c A ΔC, where k_c is the mass-transfer coefficient in m/s and ΔC the concentration difference between the surface and the bulk fluid. For drying, it is often written in terms of humidity or partial pressure differences.
Mass-transfer coefficients are correlated with dimensionless numbers analogous to those for heat transfer: the Sherwood number Sh = k_c L/D plays the role of the Nusselt number, and the Schmidt number Sc = ν/D = μ/(ρD) plays the role of the Prandtl number. Correlations such as Sh = 0.023 Re^0.83 Sc^0.33 for turbulent pipe flow parallel those for heat transfer. The analogy between heat and mass transfer, expressed by the Chilton–Colburn j-factors, allows mass-transfer coefficients to be estimated from heat-transfer data, and the Lewis number Le = α/D = Sc/Pr compares heat and mass diffusivities; for air–water vapour systems it is close to one, which simplifies psychrometric calculations.
In dairy processing, convective mass transfer governs the rate of drying of milk droplets in a spray dryer during the constant-rate period, when the surface is saturated and evaporation is limited by external transfer; the absorption of carbon dioxide or nitrogen into products; the removal of volatile off-flavours by vacuum deaeration; the dissolution of powders in water; and the cleaning of surfaces, where detergent must diffuse to and soil away from the deposit.
Membrane separations depend on mass transfer at the membrane surface. In ultrafiltration of milk, water and small solutes pass through the membrane while proteins are retained and accumulate at the surface in a boundary layer, a phenomenon called concentration polarisation. Back-diffusion of the retained solutes into the bulk is driven by the concentration difference and governed by a mass-transfer coefficient; when the surface concentration reaches a gel or limiting value, flux becomes nearly independent of pressure and is given by the film model, J = k ln(C_wall/C_bulk). Higher cross-flow velocity, raising k, increases the limiting flux, which is why membrane plants pump retentate at high velocity.
Drying combines heat and mass transfer. In the constant-rate period, the rate of evaporation equals the rate of heat transfer to the surface divided by the latent heat; in the falling-rate period, internal diffusion of water through the drying solid controls the rate, and diffusivity falls steeply as moisture falls, producing a dense skin on milk powder particles.
Oxygen transfer is a key concern in packaging and storage. Oxygen permeates packaging films at rates set by their permeability, thickness and area; it dissolves in milk and powders and drives oxidation. Products sensitive to oxidation, such as infant formula and whole milk powder, are packed in high-barrier materials under inert gas, and the oxygen transmission rate of packages is specified and tested.
Dimensionless numbers group physical quantities so that results from one system can be applied to another, and so that correlations can be written compactly. In fluid flow, the Reynolds number, Re = ρvD/μ, compares inertial and viscous forces and decides whether flow is laminar or turbulent. The Froude number, Fr = v²/(gL), compares inertial and gravitational forces and matters in free-surface flows and mixing; the power number, Po = P/(ρN³D⁵), characterises the power drawn by agitators.
In heat transfer, the Nusselt number, Nu = hL/k_f, using the fluid's conductivity, is the dimensionless heat-transfer coefficient, the ratio of convective to conductive heat transfer across the fluid layer. The Prandtl number, Pr = c_pμ/k, compares momentum and thermal diffusivity; it is about 0.7 for air, 7 for water at room temperature, and higher for milk and much higher for viscous oils and concentrates. The Grashof number, Gr = gβΔTL³/ν², characterises buoyancy-driven natural convection, and the Rayleigh number, Ra = Gr × Pr, decides whether natural convection is laminar or turbulent.
For unsteady conduction, the Biot number, Bi = hL/k_s, uses the solid's conductivity and compares internal conduction resistance with surface convection resistance; when Bi < 0.1, the solid's temperature is nearly uniform and lumped-capacity analysis applies. The Fourier number, Fo = αt/L², is dimensionless time and appears in all transient conduction charts.
Dimensional analysis organises the variables of a problem into dimensionless groups, reducing the number of independent quantities and guiding experiments. Every physical equation must be dimensionally homogeneous: each term must have the same dimensions, expressed in terms of fundamental dimensions such as mass M, length L, time T and temperature θ. Checking dimensions is a quick test of a derived formula.
The Buckingham π theorem states that if a physical relationship involves n variables containing m fundamental dimensions, it can be expressed as a relationship between n − m independent dimensionless groups, called π groups. For example, the pressure drop in a pipe depends on velocity, diameter, length, density, viscosity and roughness: seven variables including Δp, with three dimensions (M, L, T). The theorem gives 7 − 3 = 4 groups, which can be chosen as Eu = Δp/ρv², Re = ρvD/μ, L/D and ε/D. Experiments then need only find how Eu depends on the other three, rather than varying six quantities separately, and the Moody chart is exactly such a result.
The groups are found by choosing m repeating variables that together contain all the dimensions, such as ρ, v and D, and combining each remaining variable with powers of them to make a dimensionless group, solving for the exponents from the dimensional equations. Dimensional analysis does not give the form of the function, which must come from theory or experiment, and it fails if an important variable is left out.
Most engineering heat- and mass-transfer coefficients come from correlations of the form Nu = C Reᵃ Prᵇ for forced convection, Nu = C (Gr Pr)ᵃ for natural convection, and Sh = C Reᵃ Scᵇ for mass transfer, with constants determined by experiment for each geometry and flow regime. For laminar flow in pipes with a uniform wall temperature, Nu approaches 3.66 for fully developed flow, while correlations such as Sieder–Tate account for developing flow and the effect of wall temperature on viscosity, important for viscous dairy products heated in tubes. For turbulent flow, Dittus–Boelter applies to moderate temperature differences. Each correlation has a stated range of validity outside which it should not be used.
Similarity is the principle that two systems behave alike if their relevant dimensionless numbers are equal. It is the basis of scale-up from laboratory or pilot equipment to factory scale: a pilot plate heat exchanger operated at the same Reynolds and Prandtl numbers as the full-scale unit will have the same Nusselt number, so heat-transfer coefficients can be transferred. In practice it is rarely possible to match all numbers at once; for example, scaling an agitated tank at constant power per unit volume and at constant tip speed give different results, and the engineer chooses the criterion that matters most for the process, such as shear on a delicate product, blending time or heat transfer.
A mass balance applies conservation of mass to a process: for a steady process, what goes in must come out, and for a batch, input = output + accumulation. Balances are written for total mass and for each component, such as fat, protein, total solids or water, that is not created or destroyed. The method is always the same: draw a block diagram, label every stream with its mass and composition, choose a basis (such as 100 kg of feed or one hour of operation), write the balances, and solve.
Standardisation is the commonest dairy mass balance. To standardise 10,000 kg of milk at 4.2% fat to 3.5% fat by removing cream at 40% fat, let C be the cream removed and S the standardised milk: total mass, 10,000 = C + S; fat, 420 = 0.40C + 0.035S. Substituting gives 420 = 0.40C + 0.035(10,000 − C) = 350 + 0.365C, so C = 191.8 kg and S = 9,808.2 kg.
Concentration processes use the solids balance. Evaporating 10,000 kg/h of skim milk from 9% to 45% total solids: solids in = 900 kg/h, so concentrate = 900/0.45 = 2,000 kg/h, and water evaporated = 8,000 kg/h. Spray drying that concentrate to powder of 96% solids gives 900/0.96 = 937.5 kg/h of powder and removes a further 1,062.5 kg/h of water. Yield calculations for cheese, butter and powder, and reconciliation of a plant's fat and protein intake with its outputs and losses, all rest on such balances.
An energy balance applies the first law: for a steady process without work, the heat added equals the change in enthalpy of the streams, Σ(ṁh)_out − Σ(ṁh)_in = Q. Enthalpies are calculated relative to a reference temperature, with sensible heat ṁc_pΔT for liquids and latent heats for phase changes, from steam tables for water and steam.
Mixing problems use a balance with Q = 0. Mixing 2,000 kg of milk at 4 °C with 500 kg of milk at 40 °C, with equal specific heats, gives a final temperature of (2,000 × 4 + 500 × 40)/2,500 = 11.2 °C. Cooling loads use the sensible heat: cooling 10,000 L/h of milk (10,300 kg/h) from 35 °C to 4 °C requires removal of 10,300 × 3.9 × 31 = 1,245,000 kJ/h, about 346 kW, before any heat leaks or agitator work are added.
Evaporators combine mass and energy balances. In a single-effect evaporator, the steam supplied condenses and gives up its latent heat; this heat warms the feed to its boiling point and evaporates water. If 10,000 kg/h of milk enters at its boiling temperature and 8,000 kg/h of water is evaporated, with a latent heat of evaporation of about 2,350 kJ/kg at the low boiling temperature under vacuum, the heat required is about 18.8 GJ/h; steam at a latent heat of about 2,200 kJ/kg would be needed at about 8,550 kg/h, a steam economy of about 0.94 kg of water evaporated per kilogram of steam. Multiple-effect evaporators reuse the vapour of one effect to heat the next, and mechanical or thermal vapour recompression raises economy far further, which is why large dairy evaporators remove many kilograms of water per kilogram of steam.
Whole-plant analyses combine mass and energy balances to find where energy goes and where it can be saved. A milk pasteuriser balance shows the heating supplied by regeneration, hot water and steam and the cooling supplied by regeneration and chilled water; a spray dryer balance shows the energy in the inlet air, the latent heat of evaporation, the energy leaving in the exhaust air and powder, and the losses; a CIP balance shows the heat used to warm water and detergent solutions and lost in drains.
Energy use per tonne of milk processed, or per tonne of product, is a standard key performance indicator, and it varies widely with product mix: fluid milk plants use relatively little energy per tonne, while powder plants use much more, chiefly for evaporation and drying.
Balances also check data. When the measured flows and compositions do not close the mass or energy balance within the expected measurement error, a meter is wrong, a stream is missing, or there is a leak. Reconciliation of balances is therefore both an engineering design tool and a routine control of plant operation, linking this subject to plant utilities, energy management and quality assurance.
Solve every worked example yourself, then change one number and solve again until the method is automatic: Reynolds number and head loss, pump power, steam requirement from tables, conduction through a pipe wall and insulation, LMTD and regeneration efficiency, ε-NTU, and the mass balance of standardisation and evaporation. Keep units in every line of working; most examination errors are unit errors. During in-plant training, read the pasteuriser's temperature chart and calculate its regeneration efficiency from the real temperatures, then compare it with the design value.
A process or utilities engineer five years into the job sizes pipes and pumps for new lines, solves cavitation and low-flow problems, checks pasteuriser regeneration and pressure differentials, calculates steam and chilled-water loads for new products, specifies insulation, and carries out mass and energy balances to find losses. Typical decisions: whether a lobe pump is needed for a thicker dessert, whether a plate pack can take extra capacity, how much steam a new dryer will need, and which heat-recovery project pays back fastest.
An engineering manager or plant director with 25 years' experience uses these principles to judge capital projects and energy strategy: evaporator and dryer configurations, heat-pump and heat-recovery schemes, steam-system upgrades, and the trade-off between capital and running costs. Energy per tonne of product is now a measured and reported indicator, and decisions on fuel switching, electrification and heat recovery rest on sound mass and energy balances, which senior engineers must be able to check rather than accept on trust from vendors or consultants.
A professor or research engineer studies transport phenomena in dairy systems: heat transfer and fouling in UHT plants, flow and rheology of concentrates and gels, mass transfer in membrane filtration and drying, and modelling with computational fluid dynamics. Such work rests on dimensionless analysis, careful property data at process conditions, and validation against plant measurements. Teaching the fundamentals with real dairy data equips students for the advanced engineering subjects and for industrial problem-solving and for the energy transition the industry now faces.
Property values for milk and dairy products are typical; they vary with composition and temperature and should be taken from measured data for design. Compressible flow, two-phase flow, boiling heat transfer, detailed heat-exchanger mechanical design, computational fluid dynamics and pressure-vessel codes are only named; they are covered in advanced-dairy-engineering and the agricultural engineering track. Holding-tube rules for regulatory compliance differ between jurisdictions and are not quoted. Refrigeration cycles, evaporator and dryer design are treated in dairy-refrigeration-cold-chain and evaporation-drying-dairy.
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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