Food Science & Technology · Food Process Engineering · also Unit Operations I, Food Engineering I, Heat and Mass Transfer in Food Processing, Momentum Transfer
Mass and energy balances and the three transfer processes — momentum, heat and mass — applied to foods: pumping viscous products, heating cans and pouches, and moisture and gas movement through foods and packaging films.
In the exams: GATE XE-G Section 4 (mass and energy balance, momentum transfer, heat transfer, mass transfer); ICAR AIEEA-PG Code 18 Unit II
Every food process moves something: mass, momentum or heat. Milk is pumped through pipes and heat exchangers; cans are heated by steam and cooled by water; juice is concentrated and powder dried; oxygen and water vapour pass slowly through packaging films. The engineering sciences that describe these movements — material and energy balances, fluid mechanics, heat transfer and mass transfer, together called transport phenomena — are the common language of food process engineering. A technologist who can write a balance around a process, estimate the pressure drop in a line, size a heat exchanger or calculate how fast oxygen will reach a packaged food can reason quantitatively about almost any operation in a plant.
This subject teaches those tools with food examples. It starts with units and dimensional analysis, then material and energy balances (including the use of steam tables), fluid statics and the Bernoulli equation, flow of Newtonian and non-Newtonian foods in pipes, and pumps. Heat transfer follows: steady conduction, convection and the dimensionless numbers used to predict film coefficients, radiation, unsteady heating and cooling, and heat exchangers. The final chapters cover molecular diffusion, convective mass transfer and permeation through packaging films.
Several other subjects build on these foundations and refer back here. Thermal-processing-canning uses unsteady heat transfer for heat penetration into containers; evaporation-dehydration applies heat exchanger theory to boiling liquids and mass transfer to drying; chilling-freezing-frozen-foods applies conduction and convection to freezing times; food-packaging and shelf-life-stability use film permeability. Agricultural Engineering's food-dairy-engineering gives a one-subject survey of the same ground; this subject develops it fully. The topic is examined in GATE XE-G (Section 4: mass and energy balance; momentum transfer — flow rate and pressure drop relationships for Newtonian fluids in pipes and Reynolds number; heat transfer by conduction, convection and radiation, heat exchangers; mass transfer — Fick's law, convective mass transfer and permeability through single and multilayer films), in ICAR AIEEA-PG Code 18 Unit II and in engineering-oriented licensing and recruitment exams for food technologists. Questions are mostly numerical; the traps are units, the choice of correct equation and the direction of a driving force.
Engineering calculations are only as good as their units. The International System (SI) is built on seven base units, of which food engineers use mainly the metre (m), kilogram (kg), second (s), kelvin (K) and mole (mol). Derived units follow: force in newtons (1 N = 1 kg m/s²), pressure in pascals (1 Pa = 1 N/m²), energy in joules (1 J = 1 N m), power in watts (1 W = 1 J/s) and dynamic viscosity in pascal seconds (Pa s). Older units remain common in plants and handbooks: bar (10⁵ Pa), atmosphere (101.325 kPa), kilocalorie (4.184 kJ), centipoise (1 cP = 1 mPa s, the viscosity of water at about 20 °C), horsepower and British units such as °F and psi.
Three habits prevent most errors. First, convert every quantity to SI base units before substituting into an equation — kJ to J, minutes and hours to seconds, millimetres to metres, kPa to Pa. Second, carry units through each step and check that the result has the expected dimensions. Third, distinguish gauge pressure (relative to atmosphere) from absolute pressure, and temperature differences (where 1 K = 1 °C) from absolute temperatures (where kelvin must be used, for example in the gas laws and radiation).
Every physical quantity can be expressed in terms of fundamental dimensions — mass M, length L, time T and temperature θ. Velocity has dimensions L T⁻¹, force M L T⁻², pressure M L⁻¹ T⁻², viscosity M L⁻¹ T⁻¹ and thermal conductivity M L T⁻³ θ⁻¹. A physically meaningful equation must be dimensionally homogeneous: every term must have the same dimensions. This is a powerful check on algebra and on remembered formulas.
Combinations of variables in which all dimensions cancel are dimensionless groups. They are the backbone of transport phenomena because they describe the ratio of competing effects independently of scale. The Reynolds number (ρvD/μ) compares inertial and viscous forces; the Prandtl number (cp μ/k) compares momentum and thermal diffusivities; the Nusselt number (hD/k) compares convective and conductive heat transfer; the Biot number (hL/k) compares surface and internal resistances to heat transfer in a solid; the Fourier number (αt/L²) is dimensionless time for unsteady conduction; the Schmidt (μ/ρD) and Sherwood (km L/D) numbers are the mass-transfer analogues of Prandtl and Nusselt; the Grashof number describes natural convection, and the power number describes agitator power (mechanical-unit-operations).
Dimensional analysis finds the dimensionless groups that govern a problem before any experiment or theory. Buckingham's π theorem states that a relationship among n variables involving m fundamental dimensions can be rewritten as a relationship among n − m independent dimensionless groups. For example, the pressure drop ΔP in a straight pipe depends on pipe diameter D, length L, fluid velocity v, density ρ and viscosity μ — six variables in three dimensions (M, L, T) — so it can be written as a relation among three groups: ΔP/(ρv²), ρvD/μ (Reynolds number) and L/D. This is why friction factor charts can be drawn once, as functions of Reynolds number and relative roughness, and used for any fluid and pipe.
The same reasoning underlies scale-up from laboratory or pilot plant to production: if the relevant dimensionless groups are kept the same, the behaviour is the same. In practice it is rarely possible to keep all groups constant at once — for example, keeping both Reynolds number and the ratio of mixing time to residence time constant in a larger vessel — so engineers choose the group that controls the critical outcome, such as constant power per unit volume for mixing or constant Reynolds number for flow regime, and confirm with trials.
A material (mass) balance applies the law of conservation of mass to a defined system: what goes in must come out, accumulate or be consumed or generated by reaction. In general form, input + generation = output + accumulation + consumption. For steady-state continuous processes without reaction — the usual case for physical food operations — this reduces to input = output, for the total mass and for each component separately. In batch processes, the balance is written for the whole batch.
Good practice is systematic: draw a flow diagram showing every stream; label known flows and compositions; choose a basis (for example 100 kg of feed or 1 hour of operation); identify a 'tie' component that passes through unchanged, such as dry solids in drying or fat in a separation; write the total and component balances; and solve. The number of independent equations must equal the number of unknowns. Balances are used to calculate yields, to check plant data, to design recipes and to find flows that cannot be measured directly.
Mixing problems combine two or more streams to give a product of specified composition. With two streams A and B of component fractions xA and xB mixed to give a product P of fraction xP, the total balance A + B = P and the component balance A xA + B xB = P xP give A/P = (xP − xB)/(xA − xB). For example, to make 1,000 kg of juice drink at 12 °Brix by blending a 65 °Brix concentrate with water, the concentrate needed is 1,000 × 12/65 = 184.6 kg and the water 815.4 kg. The Pearson square is a graphical form of the same calculation, widely used for standardising milk fat, protein or sugar content.
When more than one component must be specified, as in formulating a product to given fat, protein and moisture from several ingredients, the balances form a set of simultaneous linear equations, solved by hand for small cases or with spreadsheets and linear programming when costs are to be minimised.
In concentration and drying, the dry solids are the tie component. For an evaporator, feed F at solids fraction xF gives concentrate P at xP, with F xF = P xP, so the water removed is W = F − P = F(1 − xF/xP); evaporating 5,000 kg/h of milk from 12 per cent to 45 per cent total solids removes 5,000 × (1 − 12/45) = 3,667 kg/h of water. For dryers, the same solids balance is applied with moisture contents converted to dry basis where convenient: 100 kg of carrots at 88 per cent moisture contain 12 kg of solids, and drying to 8 per cent moisture leaves 12/0.92 = 13.04 kg of product, so 86.96 kg of water is removed. A common slip is to subtract moisture percentages directly (88 − 8 = 80 kg), which ignores that the percentages refer to different total masses.
Balances around the air side of a dryer use the humidity ratio of the air and the dry-air flow, as shown in evaporation-dehydration. Recycle streams, bypass streams and purges — common in membrane plants, crystallisers and dryers with recirculated air — are handled by writing balances around the whole process and around individual units or mixing points.
The energy balance applies the first law of thermodynamics: energy is conserved. For a steady-flow process, energy in with streams and by heat and work equals energy out with streams. In most food-processing calculations, changes in kinetic and potential energy are negligible compared with enthalpy changes, and the balance reduces to: heat added = Σ (m h)out − Σ (m h)in, where h is specific enthalpy. Sensible heat, the heat to change temperature without phase change, is Q = m cp ΔT; latent heat, the heat for a phase change at constant temperature, is Q = m λ.
The specific heat of a food depends on its composition. Simple estimates weight the specific heats of the components: for example, Choi and Okos give temperature-dependent values for water, protein, fat, carbohydrate, fibre and ash, and the simpler Siebel equations (used in chilling-freezing-frozen-foods) use water content alone. Heating 1,000 kg of a soup with cp = 3.8 kJ/kg K from 20 to 90 °C needs 1,000 × 3.8 × 70 = 266,000 kJ. Losses to the surroundings, which may be 5 to 20 per cent in practice, are added or estimated from insulation and surface temperatures.
Steam is the main heating medium in food plants because it carries a large amount of latent heat per kilogram and releases it at a constant, pressure-controlled temperature when it condenses. Steam tables list, for saturated water and steam at each temperature or pressure, the saturation pressure or temperature, the specific volume and the specific enthalpies of saturated liquid (hf), of saturated vapour (hg) and of vaporisation (hfg = hg − hf). At atmospheric pressure (101.3 kPa), water boils at 100 °C with hfg about 2,257 kJ/kg; at about 198.5 kPa absolute (roughly 1 bar gauge) steam condenses at 120 °C with hfg about 2,202 kJ/kg. The latent heat falls as pressure and temperature rise, and becomes zero at the critical point.
Wet steam contains liquid droplets; its quality (dryness fraction) x is the mass fraction of vapour, and its enthalpy is h = hf + x hfg. Only the vapour fraction delivers latent heat, so wet steam heats less than dry saturated steam and can leave water films that reduce heat transfer; steam separators and traps are used to keep steam dry and remove condensate. Superheated steam, above the saturation temperature, gives up its superheat by convection before condensing and is less effective for direct heating; for culinary steam used in direct contact with food (steam injection, infusion, peeling), steam is filtered and made from potable water with approved boiler chemicals.
For indirect heating with condensing steam, the steam needed is the heat required divided by the latent heat released, if condensate leaves at the saturation temperature: m_steam = m cp ΔT / hfg. Heating 2,000 kg/h of milk (cp 3.9 kJ/kg K) from 10 to 72 °C with steam condensing at 120 °C (hfg 2,202 kJ/kg) needs 2,000 × 3.9 × 62/2,202 ≈ 220 kg/h of steam. If condensate is subcooled before leaving, the additional sensible heat of the condensate is also counted.
For direct steam injection, the steam condenses in the product and adds its mass: the balance must include both the steam's enthalpy and the dilution, which is often removed later by flash cooling in a vacuum vessel. For a regenerative heat exchanger in a pasteuriser, the energy balance shows how much of the heat needed is recovered from the hot pasteurised product; regeneration efficiencies of about 90 per cent or more are common in modern plants, which is why steam and cooling water demands are much lower than for heating and cooling with no recovery. Energy balances also underlie the evaporator steam calculations in evaporation-dehydration and refrigeration loads in chilling-freezing-frozen-foods.
A fluid deforms continuously under shear. Its key properties for flow are density ρ and viscosity μ, the ratio of shear stress to shear rate for Newtonian fluids. Water at 20 °C has ρ ≈ 998 kg/m³ and μ ≈ 1.0 mPa s; milk is slightly denser and about twice as viscous; honey and concentrated syrups are thousands of times more viscous, and viscosity of liquids falls sharply with rising temperature.
In a fluid at rest, pressure increases with depth: P = P0 + ρ g h. The pressure at the bottom of a 3 m column of milk (ρ = 1,030 kg/m³) exceeds that at the surface by 1,030 × 9.81 × 3 ≈ 30.3 kPa. This relation underlies manometers, which measure pressure differences as heights of liquid columns, liquid-level measurement by pressure sensors in tanks, and the hydrostatic steriliser, whose water legs balance steam pressure (thermal-processing-canning). Pressure gauges usually read gauge pressure; absolute pressure is gauge plus atmospheric.
For steady flow of an incompressible fluid, conservation of mass gives the continuity equation: the volumetric flow Q = A v is the same at every cross-section, so velocity rises where the pipe narrows: v1 A1 = v2 A2. Halving the diameter of a pipe quadruples the velocity.
The Bernoulli equation expresses conservation of mechanical energy along a streamline for an ideal, frictionless, incompressible fluid: P/ρ + v²/2 + g z = constant, or in head form, P/(ρg) + v²/(2g) + z = constant. The three terms are pressure energy, kinetic energy and potential energy per unit mass. For real flows the engineering (extended) Bernoulli equation adds energy supplied by a pump, Ep, and energy lost to friction, Ef: P1/ρ + v1²/2 + g z1 + Ep = P2/ρ + v2²/2 + g z2 + Ef. This is the basis for calculating pump power: the pump must supply the increase in pressure, velocity and elevation energy plus friction losses.
Several flow meters follow directly from the Bernoulli and continuity equations. In a Venturi meter or orifice plate, the fluid accelerates through a constriction and its pressure falls; the pressure difference is proportional to the square of the flow rate, so Q = Cd A2 √[2ΔP/(ρ(1 − (A2/A1)²))], where Cd is a discharge coefficient (close to 1 for Venturis, lower for orifices, which also cause larger permanent pressure losses). A Pitot tube measures local velocity from the difference between stagnation and static pressure, v = √(2ΔP/ρ). In food plants these devices are less used for product lines because they are difficult to clean; magnetic flow meters (for conductive liquids such as milk and juice), Coriolis mass flow meters (which also give density) and positive-displacement meters are preferred for hygienic duty and for the timing pumps of pasteurisers, where accurate flow determines holding-tube residence time. Whatever the meter, its calibration must be checked regularly, because a meter that reads low lets the line run faster than intended and shortens the actual holding time.
Osborne Reynolds showed that flow in a pipe is either laminar, with fluid moving in smooth parallel layers, or turbulent, with eddies mixing the fluid across the pipe, and that the transition depends on the dimensionless Reynolds number, Re = ρ v D / μ, where v is the mean velocity and D the internal diameter. Food engineering texts commonly take flow as laminar for Re below about 2,100, turbulent above about 4,000, and transitional between; some sources quote slightly different limits. Milk (ρ = 1,030 kg/m³, μ = 2 mPa s) flowing at 1.5 m/s in a 38 mm tube has Re = 1,030 × 1.5 × 0.038/0.002 ≈ 29,000, well into turbulent flow; honey in the same pipe would be laminar.
In laminar flow the velocity profile is parabolic, the maximum velocity at the centre is twice the mean velocity, and the fastest elements pass through a tube in half the mean residence time — important for holding tubes (thermal-processing-canning and aseptic processing). In turbulent flow the profile is much flatter, with the maximum about 1.2 times the mean, and a thin laminar sub-layer at the wall, where most of the resistance to heat and mass transfer lies.
Friction between the flowing fluid and the pipe wall causes a loss of pressure along the pipe. For laminar flow of a Newtonian fluid, the Hagen–Poiseuille equation gives ΔP = 32 μ L v / D². More generally, the pressure drop is written with a friction factor. Using the Fanning friction factor f, ΔP = 4 f (L/D) (ρ v²/2), equivalently written as 2 f ρ v² L/D; using the Darcy (Moody) friction factor fD = 4f, ΔP = fD (L/D)(ρ v²/2). Exam questions and textbooks use both, so the definition must be checked: in laminar flow, Fanning f = 16/Re and Darcy fD = 64/Re.
In turbulent flow f depends on Re and on the relative roughness ε/D of the pipe wall, and is read from the Moody chart or calculated from correlations such as the Colebrook equation. For smooth pipes, such as polished stainless-steel food tubing, the Blasius equation, Fanning f = 0.0791 Re^−0.25, is adequate for Re up to about 100,000. For the milk example above (Re ≈ 29,000), f = 0.0791 × 29,000^−0.25 ≈ 0.0060, and the pressure drop in 20 m of tube is 4 × 0.0060 × (20/0.038) × 1,030 × 1.5²/2 ≈ 14.7 kPa.
Valves, bends, tees, sudden expansions and contractions, and equipment such as filters and heat exchangers cause additional, so-called minor, losses. They are expressed as a loss coefficient K, giving a loss of K v²/2 per unit mass (or a head loss of K v²/2g), or as an equivalent length of straight pipe. Typical K values are roughly 0.3 to 0.9 for standard elbows, around 0.2 for a fully open gate valve and higher for globe and partly closed valves, 0.5 for a sharp entrance from a tank and 1.0 for exit into a tank. In short food-plant lines with many fittings, minor losses can exceed the friction losses in straight pipe.
The total energy needed to move a fluid at different flow rates, plotted as head against flow, is the system curve. It combines the static head (elevation and pressure difference, independent of flow) with friction and minor losses, which rise roughly with the square of flow in turbulent flow. The operating point of a pumping system is where the system curve crosses the pump curve, as described in the chapter on pumps.
Many liquid foods are non-Newtonian: their apparent viscosity depends on shear rate. Shear-thinning (pseudoplastic) foods, such as fruit purées, concentrated juices, sauces and many hydrocolloid solutions, become thinner the faster they are sheared; shear-thickening (dilatant) behaviour is rare in foods, seen in some concentrated starch suspensions. Some foods, such as tomato paste, mayonnaise and chocolate, have a yield stress and do not flow until a minimum stress is exceeded. The models used in engineering calculations are the power-law model, τ = K γ̇ⁿ, where K is the consistency coefficient (Pa sⁿ) and n the flow behaviour index (n < 1 shear-thinning, n = 1 Newtonian, n > 1 shear-thickening); the Bingham plastic model, τ = τ0 + μp γ̇; and the Herschel–Bulkley model, τ = τ0 + K γ̇ⁿ. The measurement of these properties is covered in food-physical-properties-rheology. For pipe-flow design the parameters must be measured over the range of shear rates found in the pipe, because extrapolating a power-law fit far outside its data can give large errors in pressure drop.
For power-law fluids in laminar pipe flow, the velocity profile is flatter than the Newtonian parabola for n < 1 and the ratio of maximum to mean velocity is (3n + 1)/(n + 1), which is 2 for Newtonian fluids and approaches 1 as n becomes small. The Reynolds number is replaced by the generalised (Metzner–Reed) Reynolds number: Re_G = ρ v^(2−n) D^n / [K 8^(n−1) ((3n + 1)/(4n))^n]. With this definition the laminar Fanning friction factor is again f = 16/Re_G, and laminar flow is commonly assumed to persist to Re_G of about 2,100, although the transition shifts somewhat with n.
The pressure drop for laminar flow of a power-law fluid can also be written directly: ΔP = 4 K L (8v/D)^n ((3n + 1)/(4n))^n / D. Because pressure drop rises with v^n rather than v, doubling the flow of a strongly shear-thinning purée (say n = 0.3) raises the laminar pressure drop by only 2^0.3 ≈ 1.23 times, while for a Newtonian liquid it doubles. Yield-stress fluids flow as a solid 'plug' at the centre of the pipe, where stress is below the yield stress, surrounded by a sheared annulus; their pumping requires a minimum pressure gradient to start flow.
Non-Newtonian behaviour affects pump selection, line sizing and residence-time calculations. Viscous and yield-stress foods usually flow laminar in plant pipes, so pressure drops are high and positive-displacement pumps are preferred. Because viscosity falls with temperature, heating a product before pumping it long distances can save energy, but it may damage quality. For holding-tube design in aseptic and HTST systems, the maximum velocity of a shear-thinning fluid is less than twice the mean, so the Newtonian factor of 2 is conservative; the actual profile must be used when the process authority accepts it. Time-dependent behaviour — thixotropy, a fall in viscosity with time of shearing, as in some yoghurts and ketchups — means that pressure drops measured at start-up may differ from steady operation. Particulate foods add further complications, because particles may move faster or slower than the carrier liquid. Where calculations are uncertain, pressure drops are measured on a test loop with the actual product, and pumps are selected with margin for the most viscous batch expected.
A pump adds mechanical energy to a liquid. From the engineering Bernoulli equation, the energy the pump must supply per unit mass is Ep = (P2 − P1)/ρ + (v2² − v1²)/2 + g(z2 − z1) + Ef, and the power delivered to the fluid is ṁ Ep. The shaft power is larger, divided by the pump efficiency η: P_shaft = ṁ Ep/η. For example, pumping 2 kg/s of juice against a total energy requirement of 300 J/kg with an efficiency of 60 per cent needs 2 × 300/0.6 = 1,000 W of shaft power. Engineers often express pump requirements as head, H = Ep/g, in metres of the liquid being pumped.
Cavitation occurs when the pressure at the pump inlet falls below the vapour pressure of the liquid, forming vapour bubbles that collapse violently inside the pump, causing noise, loss of performance and damage. To avoid it, the net positive suction head available (NPSHa) — the absolute pressure head at the inlet above the vapour pressure head — must exceed the net positive suction head required (NPSHr) given by the pump maker. Hot liquids, liquids under vacuum (as in evaporator outlets) and long, narrow suction lines are the common causes of cavitation in food plants.
Centrifugal pumps use a rotating impeller to throw liquid outward, converting kinetic energy into pressure in the casing. They are simple, compact, inexpensive, easy to clean in place and give smooth, non-pulsating flow, and they are the standard for low-viscosity liquids such as milk, water, juices, beer and cleaning solutions. Their characteristic curve shows head falling as flow rises; flow is controlled by throttling the discharge valve or, more efficiently, by variable-speed drives. The affinity laws describe how performance changes with speed N: flow is proportional to N, head to N² and power to N³. Increasing speed by 10 per cent therefore raises flow by 10 per cent, head by 21 per cent and power by about 33 per cent.
Centrifugal pumps are not self-priming in their basic form, lose efficiency rapidly as viscosity rises (above a few hundred mPa s), deliver a flow that depends on the downstream pressure, and shear the product, which can damage fat globules, curd particles or fragile pieces. They should not run against a closed valve for long, because the trapped liquid heats up.
Positive-displacement (PD) pumps trap a fixed volume of liquid and push it to the outlet, so their flow is nearly proportional to speed and largely independent of discharge pressure. Rotary lobe pumps, with two lobed rotors turning in a casing, are the most common hygienic PD pumps, handling viscous products, pieces and shear-sensitive products such as yoghurt, cream and fruit preparations. Progressing cavity (mono) pumps, gear pumps, piston pumps (as in homogenisers and fillers), diaphragm pumps and peristaltic pumps are also used. Because their flow is fixed by speed, PD pumps serve as timing pumps in HTST pasteurisers, guaranteeing the maximum flow and hence the minimum holding time. Because they can build very high pressure, a PD pump discharging against a closed valve can burst a line, so pressure relief valves or bypasses are required.
All food pumps must meet hygienic design requirements: smooth, crevice-free product-contact surfaces of suitable stainless steel and approved elastomers, self-draining casings, seals that can be cleaned and do not trap product, and suitability for cleaning-in-place. Standards such as 3-A sanitary standards and EHEDG guidelines, discussed in food-plant-design-utilities, set these requirements and are used in equipment approval.
Conduction is heat transfer 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 A dT/dx, where k is the thermal conductivity (W/m K). Metals conduct well (stainless steel about 15 W/m K, copper about 400), water poorly (about 0.6), foods between about 0.2 and 0.6 when unfrozen and higher when frozen, and insulating materials very poorly (polyurethane foam about 0.02 to 0.03). For steady conduction through a plane wall of thickness Δx, q = k A (T1 − T2)/Δx.
The analogy with electrical resistance makes complex problems simple: the thermal resistance of a plane wall is R = Δx/(kA), and heat flow equals temperature difference divided by resistance. A stainless-steel wall 2 mm thick conducting heat across 1 m² with a 10 K difference passes 15 × 1 × 10/0.002 = 75,000 W, showing why metal walls offer little resistance compared with fluid films and fouling layers.
For layers in series, as in a cold-store wall of brick, insulation and plaster, the resistances add: R_total = Σ Δxᵢ/(kᵢA), and q = (T_inside − T_outside)/R_total. Convective films at each surface add resistances 1/(hA). For example, a cold-store wall with 100 mm polyurethane (k = 0.025 W/m K) between 10 mm steel-faced panels (negligible resistance), with surface coefficients of 10 and 25 W/m² K, between −25 °C inside and 30 °C outside, has resistance per square metre 1/25 + 0.1/0.025 + 1/10 = 4.14 m² K/W, and a heat gain of 55/4.14 ≈ 13.3 W per square metre. Almost all the resistance is in the insulation.
For a cylindrical wall such as a pipe or tube, the area changes with radius, and the steady heat flow is q = 2π L k (T1 − T2)/ln(r2/r1); the resistance of a cylindrical layer is ln(r2/r1)/(2π k L). Layers of insulation on a pipe again add in series. An overall heat transfer coefficient U combines all the resistances between two fluids: 1/(U A) = Σ R. Because the area differs on the inside and outside of a tube, U must be quoted on a stated area.
Adding insulation to a small pipe increases the conduction resistance but also increases the outer surface area, reducing the outer convective resistance. Below a certain outer radius, the second effect wins and adding insulation actually increases heat loss. This critical radius for a cylinder is r_c = k/h, where k is the insulation conductivity and h the outer convective coefficient; for a sphere it is 2k/h. With k = 0.04 W/m K and h = 10 W/m² K, r_c = 4 mm, so ordinary pipes, already larger, always benefit from insulation, but very thin electrical wires can lose more heat when lightly insulated.
At interfaces between solid layers, imperfect contact creates an additional thermal contact resistance, because heat must cross small air gaps between surface roughnesses. It matters in plate freezers, where packages must be pressed firmly against plates, and in heat transfer from a heated surface to a food pressed against it, as in drum dryers and contact grills.
Convection is heat transfer between a surface and a moving fluid. It is described by Newton's law of cooling, q = h A (Ts − T∞), where h is the convective heat transfer (film) coefficient. Unlike thermal conductivity, h is not a property of the fluid alone: it depends on fluid properties, velocity, geometry and whether the flow is natural (driven by density differences from temperature) or forced (driven by pumps or fans). Typical orders of magnitude are about 5 to 25 W/m² K for natural convection in air, 10 to 200 for forced air, 50 to 1,000 for natural convection in water, 500 to 10,000 or more for forced flow of water, and 5,000 to 100,000 for condensing steam and boiling water. This range explains why steam heats so effectively, why air blast freezers are slower than plate or immersion freezers, and why the film on the product side usually controls heat exchanger performance.
Film coefficients are estimated from correlations among dimensionless numbers obtained from experiments. For forced convection the Nusselt number Nu = hD/k is expressed as a function of the Reynolds and Prandtl numbers; for natural convection, of the Grashof (or Rayleigh) and Prandtl numbers. For fully developed turbulent flow inside smooth tubes, the Dittus–Boelter equation is widely used: Nu = 0.023 Re^0.8 Pr^n, with n = 0.4 when the fluid is being heated and 0.3 when it is being cooled, valid for Re above about 10,000 and moderate Prandtl numbers; the Sieder–Tate form adds a viscosity correction for liquids whose viscosity changes much between bulk and wall. For fully developed laminar flow in a tube at constant wall temperature, Nu is constant at about 3.66, and about 4.36 at constant heat flux; short tubes with developing flow give higher values.
For example, milk with Re = 29,000 and Pr = 13 heated in a 38 mm tube (k = 0.55 W/m K) gives Nu = 0.023 × 29,000^0.8 × 13^0.4 ≈ 0.023 × 3,712 × 2.79 ≈ 238, and h = 238 × 0.55/0.038 ≈ 3,450 W/m² K. Because h rises with velocity to the power 0.8 in turbulent flow, doubling velocity raises the film coefficient by about 1.74 times, at the cost of about three to four times the pressure drop.
Every surface above absolute zero emits thermal radiation, mainly in the infrared at food-processing temperatures. A black body emits energy E = σ T⁴ per unit area, where σ = 5.67 × 10⁻⁸ W/m² K⁴ is the Stefan–Boltzmann constant and T is absolute temperature in kelvin. Real surfaces emit less, by their emissivity ε (0 to 1): polished stainless steel has low emissivity, while most foods, water, paints and oxidised metals are close to 0.9 or above. The net radiant exchange between a small grey body and large surroundings is q = ε σ A (T1⁴ − T2⁴). A food surface at 20 °C in an oven whose walls are at 200 °C, with ε = 0.9, receives 0.9 × 5.67 × 10⁻⁸ × (473⁴ − 293⁴) ≈ 2,180 W/m².
Radiation is important in baking ovens, grills, infrared heaters, roasting and freeze dryers (where it supplies part of the sublimation heat), and it contributes to heat loss from hot surfaces. Because it depends on the fourth power of absolute temperature, it becomes dominant at high temperatures. For small temperature differences it can be expressed as a radiative heat transfer coefficient added to the convective one. Microwave and radio-frequency heating are different mechanisms — volumetric heating by electromagnetic energy absorbed inside the food — and are covered in novel-nonthermal-processing.
In many food processes temperatures change with time: a can heats in a retort, a carcass cools in a chiller, a potato cooks in water. The Biot number, Bi = h L/k, compares the resistance to conduction inside the solid (L/k) with the resistance to convection at its surface (1/h), where L is a characteristic length. If Bi is small (commonly below 0.1, with L taken as volume/surface area), internal resistance is negligible: the solid is at nearly uniform temperature, and its temperature follows the lumped-capacitance solution, (T − T∞)/(Ti − T∞) = exp(−h A t/(ρ cp V)). The group ρ cp V/(h A) is a time constant: after one time constant the temperature difference has fallen to 1/e (37 per cent) of its initial value.
Lumped analysis suits small pieces with high conductivity or low surface coefficients — peas in air, thin metal containers of well-stirred liquid, thermocouples. For example, a small sphere of food 1 cm in diameter (V/A = r/3 = 1.67 mm) with k = 0.5 W/m K in air with h = 20 W/m² K has Bi = 20 × 0.00167/0.5 ≈ 0.067, so lumped analysis applies.
When the Biot number is larger, temperature varies within the solid, and the heat conduction equation must be solved with time. For simple shapes — infinite slab, infinite cylinder and sphere — with constant properties and a sudden change of surrounding temperature, analytical solutions exist as infinite series. They depend on the Biot number (here using the half-thickness or radius) and on the Fourier number, Fo = α t/L², where α = k/(ρ cp) is the thermal diffusivity. Foods have thermal diffusivities of about 1.2 to 1.6 × 10⁻⁷ m²/s, close to that of water — very low, which is why large foods heat and cool slowly.
Heisler charts plot the solutions: for a given Bi, the dimensionless centre temperature (T − T∞)/(Ti − T∞) is read against Fo, and supplementary charts give temperatures at other positions. For Fo greater than about 0.2, the first term of the series is sufficient, and the logarithm of the dimensionless temperature decreases linearly with time — the basis of the straight-line heating curves (fh and j) used in thermal-processing-canning. A consequence of the Fourier number is that heating time scales with the square of size: a can or piece twice as thick takes about four times as long to reach the same centre temperature when internal conduction controls.
Real foods and containers are finite: a can is a short cylinder, a carton a rectangular block. For such shapes the dimensionless temperature can be found as the product of the solutions for the intersecting infinite shapes: for a finite cylinder, the product of the infinite-cylinder and infinite-slab solutions; for a brick, the product of three slab solutions (Newman's rule). This is how centre temperatures of cans and cartons are estimated by hand. Irregular shapes, temperature-dependent properties (especially during freezing and thawing), internal heat generation (respiration, microwave heating) and changing boundary conditions are handled by numerical methods — finite differences or finite elements — now routine in food engineering software. Unsteady heat transfer is applied in thermal-processing-canning (heat penetration), chilling-freezing-frozen-foods (cooling and freezing times) and in cooking, baking and pasteurisation in package. Whatever the method, predicted temperatures are checked against thermocouple measurements at the slowest-heating point before a process is relied on.
Heat exchangers transfer heat between two fluids separated by a wall. Plate heat exchangers (PHEs) consist of corrugated stainless-steel plates clamped in a frame, with gaskets directing the two fluids through alternate channels. The corrugations create turbulence at low Reynolds numbers, giving high film coefficients, compact size and easy inspection and expansion; PHEs dominate milk, juice and beer pasteurisation, with heating, regeneration and cooling sections in one frame. Their limits are pressure (gaskets), and products containing particles or fibres that can block narrow channels.
Tubular heat exchangers — concentric tube (double pipe), triple tube and shell-and-tube designs — tolerate higher pressures and products with particles or higher viscosity, and are used in UHT plants, for pulpy juices and for sauces. Scraped-surface heat exchangers (SSHEs) have a rotating shaft with blades that continuously scrape the product film from the heated or cooled cylinder wall, so they can heat, cool or freeze very viscous, sticky or particulate products — purées, sauces with pieces, margarine and ice-cream mixes — though at higher cost and lower capacity. Direct heating by steam injection or infusion, used in UHT processing, transfers heat with no wall at all.
The heat transferred is Q = U A ΔT_m, where U is the overall heat transfer coefficient, A the area and ΔT_m the mean temperature difference. U is found from the resistances in series: 1/U = 1/h_i + Δx/k_w + 1/h_o + R_f (for a thin wall, ignoring area differences), where R_f is the fouling resistance. Because the smallest film coefficient dominates, improving the product-side coefficient usually matters most; fouling by milk proteins and minerals can halve U during a run.
For flow in which both fluids change temperature, the appropriate mean is the log mean temperature difference, LMTD = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2), where ΔT1 and ΔT2 are the temperature differences at the two ends. For example, milk heated from 10 to 60 °C by hot water cooling from 80 to 40 °C in counter-current flow has ΔT1 = 80 − 60 = 20 K and ΔT2 = 40 − 10 = 30 K, so LMTD = (30 − 20)/ln(30/20) = 24.7 K. In co-current flow with the same terminal temperatures this arrangement is impossible, because the cold fluid would leave hotter than the hot fluid; in general counter-current flow gives a larger LMTD and allows the cold fluid to leave hotter than the hot fluid's outlet. For multi-pass and cross-flow arrangements, the counter-current LMTD is multiplied by a correction factor F (less than 1).
When outlet temperatures are unknown — for example, when checking how an existing exchanger will perform with a new flow — the effectiveness–NTU method is more convenient. The heat capacity rates are C = ṁ cp for each fluid; Cmin is the smaller. The maximum possible heat transfer is Qmax = Cmin (Th,in − Tc,in), and the effectiveness ε = Q/Qmax. The number of transfer units NTU = UA/Cmin. For a counter-current exchanger, ε = [1 − exp(−NTU(1 − Cr))]/[1 − Cr exp(−NTU(1 − Cr))], where Cr = Cmin/Cmax; when Cr = 1, ε = NTU/(1 + NTU). Charts and formulas exist for other arrangements.
In pasteurisers, the regeneration section uses hot pasteurised milk to preheat incoming cold milk. With equal flows and heat capacities (Cr = 1), regeneration efficiency is the fraction of the heating duty supplied by regeneration: (temperature rise of cold milk in the regenerator)/(pasteurisation temperature − raw milk temperature). Raising raw milk from 4 to 65 °C in a regenerator before final heating to 72 °C gives (65 − 4)/(72 − 4) ≈ 90 per cent regeneration, so only the last 7 K is supplied by hot water and a similar small cooling duty remains. A pressure differential is maintained so that pasteurised milk is always at higher pressure than raw milk in the regenerator, so that any pinhole leak flows from pasteurised to raw rather than the reverse.
Mass transfer is the movement of a component from a region of higher to lower concentration (more precisely, chemical potential). Molecular diffusion is the movement caused by random molecular motion. Fick's first law states that the diffusive flux is proportional to the concentration gradient: J = −D dC/dx, where D is the diffusion coefficient (diffusivity, m²/s). Diffusivities are of the order of 10⁻⁵ m²/s in gases, 10⁻⁹ m²/s in liquids and much smaller in solids and foods — often 10⁻¹⁰ to 10⁻¹² m²/s for water and solutes in food matrices, where they are called effective diffusivities. For steady diffusion through a layer of thickness L, the flux is J = D (C1 − C2)/L.
Fick's second law describes unsteady diffusion: ∂C/∂t = D ∂²C/∂x². Its solutions have the same form as those for unsteady heat conduction, with D replacing thermal diffusivity and a mass-transfer Fourier number Dt/L². This analogy is used for moisture movement in drying (evaporation-dehydration), salt uptake in brining and curing, sugar uptake in osmotic dehydration and candying, and loss of solutes during blanching. As with heat conduction, the time to reach a given degree of equilibration scales with the square of thickness.
At the interface between a food and a moving fluid, mass transfer is described, by analogy with Newton's law of cooling, as N = km A (Cs − C∞), where km is the convective mass transfer coefficient (m/s). Correlations for km use the Sherwood number Sh = km L/D, the Reynolds number and the Schmidt number Sc = μ/(ρD), in the same forms as heat transfer correlations with Nu and Pr. The Chilton–Colburn analogy relates heat and mass transfer: when both occur under similar flow conditions, the heat-transfer coefficient and mass-transfer coefficient are linked through the ratio of the Schmidt and Prandtl numbers, and for air–water systems the Lewis relation (h/(km ρ cp) ≈ 1 for air–water vapour) underlies the wet-bulb temperature.
Convective mass transfer controls the constant-rate period of drying, evaporation of water from cooling carcasses (weight loss), uptake of oxygen at a food surface, dissolution of ingredients, and the rate at which sublimation occurs from unpackaged frozen foods. Increasing air velocity raises km, just as it raises h.
Packaging films and coatings are not perfect barriers: gases, water vapour and aroma compounds permeate through them by a solution–diffusion mechanism. The permeant dissolves in the film at the high-pressure face, diffuses through the polymer, and desorbs at the low-pressure face. If solubility follows Henry's law (concentration S × p, where S is the solubility coefficient), the steady flux through a film of thickness L is J = D S (p1 − p2)/L = P (p1 − p2)/L, where P = D × S is the permeability coefficient. The amount transmitted in time t through area A is then q = P A t Δp/L. Permeability depends on the polymer and permeant and increases with temperature; for many hydrophilic films, such as ethylene vinyl alcohol, oxygen permeability rises sharply with humidity.
Multilayer laminates and coextrusions combine materials with complementary properties — for example a polyolefin that blocks water vapour with an ethylene vinyl alcohol or polyamide layer that blocks oxygen. Layers act as resistances in series: the overall permeance is found from L_total/P_eff = Σ Lᵢ/Pᵢ, or equivalently the total resistance is the sum of Lᵢ/(Pᵢ A). The layer with the lowest permeance per thickness dominates. For example, a laminate of 25 µm of a film with oxygen permeability P1 and 15 µm of a barrier layer with permeability one thousand times lower will transmit oxygen at nearly the rate set by the barrier layer alone. Packaging tests express performance as water vapour transmission rate (WVTR, g/m² day at stated temperature and humidity) and oxygen transmission rate (OTR, cm³/m² day at stated conditions). These values, combined with sorption isotherms or oxygen-sensitivity data, are used to predict shelf life in shelf-life-stability and food-packaging.
A student uses this subject for the calculations in GATE Section 4 and ICAR Unit II: material and energy balances, Reynolds number and pressure drop, pump power, conduction through composite walls, film coefficients from Dittus–Boelter, LMTD and effectiveness, Biot and Fourier numbers, and permeation through packaging films. In the laboratory, the same student measures pressure drop in a pipe loop, determines U for a plate heat exchanger, records cooling curves of food pieces and compares them with lumped and Heisler predictions.
A process engineer five years into a dairy, beverage or ready-meal plant uses these tools daily: checking that a new product can be pumped with existing pumps and lines, estimating steam and cooling-water needs for a recipe change, diagnosing why a pasteuriser's regeneration has fallen (fouling, gasket bypass), setting holding-tube flows from timing-pump speed, sizing an extra heat-exchanger section, and estimating whether a cheaper film will keep a snack crisp for its shelf life. They also check suppliers' claims for equipment performance against simple balances before purchase.
A senior engineer or technical director uses transport phenomena to judge capital projects and energy programmes: heat recovery across a plant, choice between plate, tubular and scraped-surface exchangers for a new product range, insulation and refrigeration loads for a new cold store, and the scale-up of pilot processes to production. They review consultants' calculations, set design standards and train younger engineers to check results with balances and dimensionless numbers. They also answer for energy and water use per tonne of product to management and auditors.
A research leader extends these foundations to coupled heat, mass and momentum transfer in complex foods — computational fluid dynamics of mixing, heating and cooling, multiphase flow of particulate foods in aseptic systems, mass transfer in porous media during drying and frying, and transport through active and biodegradable packaging. They develop and validate models used by industry and regulators, supervise doctoral research and write the examination questions that test understanding of first principles. They also teach how to judge when a simple correlation is good enough and when a full model is needed.
This manual does not reproduce steam tables, the Moody chart or Heisler charts (questions supply the values they need), nor full derivations of the heat conduction and diffusion equations. Two-phase flow, compressible gas flow, boiling and condensation correlations, and computational fluid dynamics are mentioned only briefly. Rheological measurement is in food-physical-properties-rheology; hygienic design standards for pumps and exchangers in food-plant-design-utilities; package shelf-life calculations in shelf-life-stability and food-packaging. Typical property values quoted (conductivities, diffusivities, film coefficients, loss coefficients) are orders of magnitude for orientation, not design data.
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