Agricultural Engineering · Engineering Sciences · also Thermodynamics and Heat Transfer, Unit Operations
Conduction, convection, radiation, heat exchangers and diffusion — the physics of drying, cooling, freezing and cold-store walls.
In the exams: GATE AG Section 7 (heat exchangers and composite walls each asked in 7 of 8 papers 2019–26); ICAR AICE subject 41 Unit 2; NCEES PE processing
Heat and mass transfer is the study of how thermal energy and chemical species move from one place to another because of a difference in temperature or concentration. Thermodynamics tells an engineer how much energy a process needs; heat transfer tells the engineer how fast that energy moves and how large a wall, a pipe, a fin or a heat exchanger must be to move it. Mass transfer does the same for water vapour, gases and solutes. The three modes of heat transfer are conduction (energy passed between molecules in a solid or a still fluid), convection (conduction assisted by bulk fluid motion) and radiation (electromagnetic energy exchanged between surfaces without any medium). The two modes of mass transfer are molecular diffusion and convective mass transfer.
In agricultural and food engineering almost every process is a heat and mass transfer problem. A cold-store wall is a composite conduction problem. A milk pasteuriser, an ammonia condenser and a biogas-plant heater are heat exchangers. Blanching a vegetable or chilling a carcass is transient conduction. A grain dryer, a solar dryer and a spray dryer move heat into the product while water leaves it: simultaneous heat and mass transfer. Evaporative coolers for produce and livestock rely on the same physics, and so does moisture loss through packaging films.
The subject is examined worldwide as a calculation subject. In India, GATE Agricultural Engineering places it under Dairy and Food Engineering, and composite-wall and heat-exchanger problems have appeared in most recent papers; ICAR AIEEA-PG and ASRB NET ask one-line conceptual MCQs; the ICAR AICE-JRF/SRF paper for Processing and Food Engineering expects PG-level depth, including transient conduction and drying models. In the USA the NCEES FE exam has a Thermodynamics and Heat Transfer block, and the PE Agricultural and Biological exam applies the material to processing systems and facilities. The Philippine ABE licensure, Bangladesh BCS and the Engineers Canada agricultural syllabus ('heat engineering') cover the same core. This manual follows the order of a standard textbook: conduction, fins, transient conduction, convection, phase change, radiation, heat exchangers, then diffusion, convective mass transfer and drying.
Every numerical answer in this subject depends on consistent units. Work in SI: watts, metres, kelvin (or degrees Celsius for temperature differences, which are the same size), W/m·K for conductivity and W/m²·K for heat-transfer coefficients. Radiation calculations must use absolute temperature in kelvin. Property values such as conductivity, viscosity and diffusivity depend on temperature and material, so in practice they are always read from tables for the case at hand; the values used in examples here are only illustrative.
Conduction is described by Fourier's law: the heat flux q (W/m²) in a direction x equals −k dT/dx, where k is the thermal conductivity of the material in W/m·K. The minus sign says heat flows down the temperature gradient. Conductivity is a material property and changes with temperature. Order-of-magnitude values worth remembering are: pure metals such as copper and aluminium, a few hundred W/m·K; carbon steel, tens of W/m·K; stainless steel, about 15 W/m·K; water, about 0.6 W/m·K; still air, about 0.026 W/m·K; and insulating materials such as mineral wool, expanded polystyrene and polyurethane foam, a few hundredths of a W/m·K. Insulations work because they trap still air or gas in small cells, so convection inside them is suppressed. Moist or wet insulation conducts far more heat than dry insulation, which is why cold-store panels need a vapour barrier. Two related properties appear throughout the subject. Thermal diffusivity, α = k/(ρ c), in m²/s, measures how fast a temperature change spreads through a material; a high-diffusivity material responds quickly. Specific heat c, in J/kg·K, measures how much energy a kilogram stores per degree. Foods with high water content have conductivities of roughly 0.4–0.6 W/m·K and specific heats close to that of water, and both fall as water content falls, but actual values must be taken from property tables or measured for the product in question.
For steady one-dimensional conduction with no heat generation and constant k, the temperature profile through a plane wall is a straight line and the heat flow is Q = k A (T1 − T2)/L, where A is the area normal to the flow and L the thickness. This has the same form as Ohm's law, I = ΔV/R, so an engineer writes Q = ΔT/R with the conduction resistance R = L/(k A) in K/W. Convection at a surface has the resistance 1/(h A), where h is the convective heat-transfer coefficient, and radiation can be treated the same way once it is linearised. The resistance idea is the most useful tool in the subject because resistances in series add, and resistances in parallel combine like electrical resistors in parallel. If conductivity varies linearly with temperature, k = k0(1 + βT), the heat flow for a plane wall is found by using the arithmetic-mean conductivity evaluated at the mean temperature of the two faces, and the temperature profile becomes curved rather than straight.
Pipes, tubes, tanks and insulated steam lines are cylinders. In a cylinder the area through which heat flows grows with radius, so the temperature profile is logarithmic, not linear. For a hollow cylinder of inner radius r1, outer radius r2 and length L, the conduction resistance is R = ln(r2/r1)/(2π k L) and Q = 2π k L (T1 − T2)/ln(r2/r1). For a hollow sphere the resistance is R = (r2 − r1)/(4π k r1 r2). A common mistake is to use the plane-wall formula with the thickness r2 − r1; this is acceptable only when the wall is thin compared with the radius, and then the mean area should be used. For a cylinder the correct mean area that makes the plane-wall formula exact is the logarithmic-mean area, A_lm = (A2 − A1)/ln(A2/A1); for a sphere it is the geometric mean, √(A1 A2). When a pipe carries a hot fluid inside and loses heat to air outside, the total resistance per unit length is 1/(h_i 2π r1) + ln(r2/r1)/(2π k_pipe) + ln(r3/r2)/(2π k_ins) + 1/(h_o 2π r3), and the heat loss per metre is the overall temperature difference divided by this sum.
Some bodies generate heat internally: an electric heating element, a nuclear fuel rod, a fermenting silage mass or a pile of respiring fruit. For a plane wall of thickness 2L with uniform volumetric generation q''' (W/m³) and both faces held at Ts, the temperature profile is parabolic and the maximum temperature at the centre is Ts + q''' L²/(2k). For a solid cylinder of radius R it is Ts + q''' R²/(4k), and for a sphere Ts + q''' R²/(6k). These results explain why large heaps of respiring produce or damp hay can overheat at the centre: generated heat must be conducted a long distance through a poorly conducting mass, so the central temperature rise grows with the square of the size. Doubling the depth of a heap multiplies the centre temperature rise by four, all else equal. The same reasoning is behind the advice to store respiring produce in ventilated crates rather than solid piles, and to keep bulk hay and silage below the moisture at which microbial heating becomes vigorous. In each case the remedy is either to reduce the conduction path or to carry heat away by forced airflow.
A cold-store wall is typically a sandwich: an outer skin of brick, concrete or steel sheet, a thick layer of insulation, a vapour barrier and an inner lining. For layers in series the same heat flow passes through each, so the total resistance is the sum: R_total = 1/(h_o A) + L1/(k1 A) + L2/(k2 A) + L3/(k3 A) + 1/(h_i A). The heat flow is Q = (T_outside − T_inside)/R_total, and the overall heat-transfer coefficient is U = 1/(A R_total). Once Q is known, the temperature at any interface is found by walking through the resistances: T at the first interface = T_outside − Q (1/(h_o A) + L1/(k1 A)), and so on. The largest temperature drop always occurs across the largest resistance, which in a well-designed wall is the insulation. When a wall contains elements side by side, such as timber studs between insulation batts, the two paths are in parallel and their conductances add; the parallel-path model assumes the paths do not exchange heat sideways, which is an approximation. A few timber or steel members bridging an insulation layer can raise the heat gain noticeably; these 'thermal bridges' are a practical reason why sandwich panels with continuous foam cores are preferred for cold rooms.
When two solid surfaces are pressed together, they touch only at high points of their roughness, and the gaps are filled with air. This adds a thermal contact resistance at the interface, usually quoted per unit area in m²·K/W. It falls as contact pressure rises, as the surfaces become smoother and softer, and when the gap is filled with a conducting grease or paste. In heat exchangers, a similar extra resistance develops in service as deposits build up on the surfaces: milk-stone in dairy plate heat exchangers, scale in boilers and biofilm in water circuits. This fouling resistance is added in series to the clean resistances when the exchanger is designed, which is why a new exchanger is usually slightly oversized. Fouling factors are taken from design codes or experience for the fluid pair, and cleaning schedules (such as clean-in-place cycles in a dairy) are set partly by how fast U falls. In a plate pasteuriser a steady drop in outlet temperature at constant heating-medium temperature is a sign that fouling is building up. A given fouling resistance matters most in exchangers whose clean resistances are small (high U), where it can become the controlling resistance.
Adding insulation to a flat wall always reduces heat loss. Adding insulation to a small pipe or wire does not necessarily do so, because the insulation increases the outer surface area and so reduces the outer convection resistance even while it adds conduction resistance. For a cylinder with insulation of conductivity k and outer convection coefficient h, the total resistance per unit length, ln(r/r1)/(2πk) + 1/(2π r h), has a minimum at r = k/h. This radius is the critical radius of insulation, r_c = k/h. For a sphere it is r_c = 2k/h. If the bare outer radius is less than r_c, adding insulation increases heat loss until the outer radius reaches r_c; heat loss then falls, and only after the radius passes a larger value does the insulated pipe lose less heat than the bare pipe. If the bare radius is already larger than r_c, any insulation reduces heat loss. With typical insulations (k about 0.04 W/m·K) and natural-convection coefficients (h about 5–10 W/m²·K), r_c is a few millimetres, so steam and refrigerant pipes are always larger than r_c and insulation helps. The effect is used deliberately with electric cables: a plastic sheath on a thin wire can increase heat dissipation and keep the conductor cooler.
When the convection coefficient on one side of a surface is low, as with air, the surface area on that side can be increased with fins: the ribs on an air-cooled engine cylinder, the plate fins on a refrigerator condenser and evaporator coil, and the fins on a solar-air-heater absorber. A fin conducts heat along its length and loses it by convection along its sides, so its temperature falls towards the tip. For a fin of uniform cross-section A_c, perimeter P, conductivity k and surface coefficient h, the excess temperature θ = T − T∞ obeys d²θ/dx² − m²θ = 0 with m = √(hP/(kA_c)). The solution depends on the tip condition. For a very long fin (infinite fin), θ/θ_b = e^(−mx) and the heat flow is Q = √(hPkA_c) θ_b. For a fin with an insulated (adiabatic) tip, Q = √(hPkA_c) θ_b tanh(mL). For a fin whose tip also convects, the insulated-tip formula can still be used if the length is replaced by a corrected length: L_c = L + t/2 for a thin rectangular fin of thickness t, or L_c = L + D/4 for a pin fin of diameter D. For a thin rectangular fin, P ≈ 2w and A_c = wt, so m ≈ √(2h/(kt)). Because tanh(mL) is already about 0.99 when mL = 2.65, a fin longer than about mL = 3 adds little: the extra length is almost at ambient temperature.
Two performance measures are used and they are often confused. Fin efficiency, η_f, is the actual heat flow divided by the heat flow the fin would give if its whole surface were at the base temperature. For an insulated-tip fin of uniform section, η_f = tanh(mL)/(mL). It is always less than one, it falls as the fin gets longer, and it is higher for high-conductivity, thick fins in low-h surroundings. Fin effectiveness, ε_f, is the heat flow with the fin divided by the heat flow from the same base area without the fin. For an infinite fin it equals √(kP/(hA_c)). Effectiveness should be well above one to justify a fin; a common design rule is ε_f of at least about 2. The formulas show when fins are worth using: high k (aluminium or copper), thin closely spaced fins (large P/A_c) and, above all, a low h. Fins in air or other gases pay; fins in boiling or condensing liquids with very high h rarely do. For a finned surface with total fin area A_f and unfinned base area A_b, the heat flow is Q = h (A_b + η_f A_f) θ_b, and the overall surface efficiency is η_o = 1 − (A_f/A)(1 − η_f), where A is the total area.
In transient (unsteady) conduction temperatures change with time: a can in a retort, an egg being cooled, a carcass in a chiller, a fruit in a hydrocooler. The first question is whether temperature inside the body stays nearly uniform. The Biot number answers it: Bi = h L_c/k, where L_c = V/A_s is the ratio of volume to surface area. Bi compares the internal conduction resistance with the external convection resistance. If Bi < 0.1, internal gradients are small (within about 5%) and the body can be treated as a single lump. An energy balance on the lump gives ρ V c dT/dt = −h A_s (T − T∞), whose solution is (T − T∞)/(T_i − T∞) = exp(−t/τ), with time constant τ = ρ V c/(h A_s). After one time constant the excess temperature has fallen to 36.8% of its initial value, and after about four to five time constants the body is within 2% or less of the surroundings. The same model explains the response time of thermocouples and temperature probes: a fine wire bead has a small V/A_s and responds quickly. For a sphere, L_c = r/3; for a long cylinder, L_c = r/2; for a plane wall cooled on both faces, L_c equals the half-thickness. The exponent can also be written as Bi × Fo, where the Fourier number Fo = α t/L_c² is dimensionless time.
When Bi exceeds 0.1, as for most whole fruits and vegetables in cold air or water, the centre lags behind the surface and the lumped model underestimates cooling time. The exact solutions for an infinite slab, an infinite cylinder and a sphere suddenly exposed to a fluid are infinite series. For Fo greater than about 0.2 all terms except the first are negligible, and the one-term approximation θ_o/θ_i = C1 exp(−ζ1² Fo) is accurate. The Heisler charts plot this centre temperature against Fo with 1/Bi as the parameter; companion charts give the temperature at other positions as a fraction of the centre value and the fraction of the maximum possible heat transferred, Q/Q_o (the Gröber charts). In these charts the characteristic length is the half-thickness of the slab or the outer radius of the cylinder or sphere, not V/A_s. Reading a chart: compute Bi = hL/k and Fo = αt/L², find the curve for 1/Bi, and read θ_o/θ_i. For shapes such as a short cylinder or a rectangular block the product solution applies: the dimensionless temperature equals the product of the dimensionless temperatures of the intersecting one-dimensional bodies. In produce cooling the half-cooling time and seven-eighths cooling time are used in practice; in the one-term region each successive halving of the excess temperature takes the same time.
When heat has penetrated only a short distance into a thick body, the body behaves as if it were infinitely deep. Soil heated by the sun, the surface of a thick concrete floor and the early stage of heating any large body fit this model. For a sudden change in surface temperature from T_i to T_s, (T − T_s)/(T_i − T_s) = erf(x/(2√(αt))), where erf is the error function. Because erf(2) ≈ 0.995, the depth affected is roughly 4√(αt); penetration depth grows with the square root of time. The soil application is important in agriculture: with a periodic surface temperature, the amplitude of the daily or annual wave decays exponentially with depth, and deeper soil lags the surface in time. This is why buried pipes and underground stores show small temperature swings. The damping depth for a periodic wave is √(αP/π), where P is the period, so the annual wave penetrates about √365 ≈ 19 times deeper than the daily wave in the same soil. Soil diffusivity depends strongly on moisture and compaction, so field values must be measured or taken from local data. The semi-infinite model also gives the surface heat flux, which is highest at the instant the surface temperature changes and falls as 1/√t afterwards.
Convective heat flow from a surface is written with Newton's law of cooling: Q = h A (T_s − T∞). The heat-transfer coefficient h is not a property of the fluid; it depends on the fluid, its velocity, the geometry and whether the flow is laminar or turbulent. Typical ranges (Incropera) are: free convection in gases 2–25 W/m²·K, free convection in liquids 50–1,000, forced convection in gases 25–250, forced convection in liquids 100–20,000, and boiling or condensation 2,500–100,000. Next to any surface a fluid comes to rest (the no-slip condition), so heat passes through the innermost layer by conduction. The velocity boundary layer is the region where velocity rises from zero to the free-stream value; the thermal boundary layer is the region where temperature changes from the wall value to the free-stream value. On a flat plate the laminar velocity boundary layer grows as δ ≈ 5x/√Re_x, and the ratio of velocity to thermal boundary-layer thickness is about Pr^(1/3) for laminar flow. Flow along a flat plate usually becomes turbulent at a local Reynolds number of about 5 × 10⁵; flow in a circular tube is laminar below about Re = 2,300.
Convection correlations are written in dimensionless groups so that one experiment can serve many fluids and sizes. The Reynolds number, Re = ρVL/μ = VL/ν, is the ratio of inertia to viscous forces and decides laminar or turbulent flow. The Prandtl number, Pr = ν/α = c_p μ/k, is a fluid property comparing momentum diffusivity with thermal diffusivity; it is about 0.7 for air, about 7 for water near room temperature, and much higher for oils and viscous food liquids. The Nusselt number, Nu = hL/k_fluid, is the dimensionless heat-transfer coefficient: the ratio of convective to purely conductive heat transfer across the fluid layer. Note that k in Nu is the conductivity of the fluid, whereas k in the Biot number is that of the solid. The Grashof number, Gr = gβΔT L³/ν², is the ratio of buoyancy to viscous forces and replaces Re in free convection; the Rayleigh number is Ra = Gr × Pr. The Stanton number, St = h/(ρ V c_p) = Nu/(Re Pr), and the Peclet number, Pe = Re × Pr, also appear. For free convection Nu = f(Gr, Pr) or f(Ra); for forced convection Nu = f(Re, Pr); in mixed convection the ratio Gr/Re² indicates which effect dominates.
The correlations below are the ones examinations expect; others exist for special geometries. Laminar flow over a flat plate at uniform temperature: local Nu_x = 0.332 Re_x^0.5 Pr^(1/3) and average Nu_L = 0.664 Re_L^0.5 Pr^(1/3), valid for Pr ≥ 0.6; the average coefficient over a length L is twice the local value at x = L. Fully developed turbulent flow in a smooth tube: the Dittus–Boelter equation Nu = 0.023 Re^0.8 Pr^n with n = 0.4 when the fluid is being heated and n = 0.3 when it is being cooled, valid for Re above about 10,000, 0.6 ≤ Pr ≤ 160 and L/D above about 10. Fully developed laminar flow in a circular tube: Nu = 3.66 for uniform wall temperature and 4.36 for uniform heat flux, independent of Re. Free convection from a vertical plate (McAdams-type): Nu = 0.59 Ra^(1/4) for 10⁴ < Ra < 10⁹ (laminar) and Nu = 0.10 Ra^(1/3) for 10⁹ < Ra < 10¹³ (turbulent); in the turbulent range h does not depend on the plate height. Properties are evaluated at the film temperature, the mean of the surface and free-stream temperatures, for external flow, and at the bulk mean temperature for internal flow. Two consequences are worth remembering. In turbulent tube flow h varies as V^0.8/D^0.2, so doubling velocity raises h by about 74%. In laminar flow over a plate h varies as √(V/L). The Reynolds analogy, St = C_f/2, and the Chilton–Colburn analogy, St Pr^(2/3) = C_f/2 (the j-factor), link heat transfer to friction and, in Chapter 11, to mass transfer.
When vapour touches a surface colder than its saturation temperature it condenses. If the liquid wets the surface it forms a continuous film (filmwise condensation); the film is the main resistance to heat flow, and it thickens as it runs down the surface. If the surface is not wetted, for example because of a promoter coating, the liquid forms drops that grow and roll off, exposing bare metal (dropwise condensation); coefficients are then often around an order of magnitude higher, but dropwise condensation is hard to sustain in industrial equipment, so condensers are designed for filmwise condensation. Nusselt's analysis of laminar film condensation on a vertical plate of height L gives the average coefficient h = 0.943 [ρ_l(ρ_l − ρ_v) g h_fg k_l³ / (μ_l L (T_sat − T_w))]^(1/4). For a single horizontal tube of diameter D the constant becomes about 0.73 and D replaces L. Because h varies as L^(−1/4) or D^(−1/4), a horizontal tube is usually better than a vertical tube of the same size, which is why steam condensers and refrigeration condensers are commonly built with horizontal tubes. In a vertical bank of horizontal tubes the condensate from upper tubes falls on lower tubes and thickens their film, so the average coefficient falls as the number of tubes in a column rises. Non-condensable gases such as air accumulate near the surface and sharply reduce the condensation rate; steam heaters, retorts and refrigerant condensers are therefore vented or purged. In a retort, residual air also causes cold spots, which is a food-safety issue as well as a heat-transfer one.
Boiling takes place when a surface is hotter than the saturation temperature of the liquid touching it. The boiling curve, first measured for water by Nukiyama, plots heat flux against excess temperature ΔT_e = T_w − T_sat and has four regimes. At small ΔT_e, heat moves by natural convection and vapour forms at the free surface. As ΔT_e rises, bubbles form at nucleation sites on the surface (nucleate boiling); bubble agitation gives very high coefficients, and the flux rises steeply. The flux peaks at the critical heat flux. Beyond it, in transition boiling, patches of vapour blanket the surface and the flux falls with rising ΔT_e, reaching a minimum at the Leidenfrost point. At still higher ΔT_e a stable vapour film covers the surface (film boiling), and radiation across the film becomes important. If a heater is controlled by heat input, such as an electric element, raising the input above the critical heat flux makes the surface temperature jump to the film-boiling branch, which may exceed the melting point of the heater; this 'burnout' is why equipment is designed to stay in the nucleate region with a margin below the critical flux. Nucleate boiling is usually correlated by the Rohsenow equation, and the critical heat flux by the Zuber–Kutateladze relation; both depend on the liquid and the surface condition, so values must be taken from the correlation for the case at hand. In food and dairy engineering these ideas govern evaporators for milk and juice concentration, steam jackets on kettles and the steam side of pasteurisers.
Every surface above absolute zero emits electromagnetic radiation. Thermal radiation lies roughly between 0.1 and 100 µm, covering part of the ultraviolet, the visible band and the infrared. A black body is an ideal surface that absorbs all incident radiation and, at a given temperature, emits the maximum possible energy at every wavelength. Its total emissive power is given by the Stefan–Boltzmann law, E_b = σT⁴, with σ = 5.67 × 10⁻⁸ W/m²·K⁴ and T in kelvin. Doubling absolute temperature raises emission sixteen-fold. The spectral distribution follows Planck's law, and its peak moves to shorter wavelengths as temperature rises according to Wien's displacement law, λ_max T ≈ 2,898 µm·K. The sun, at an effective temperature of about 5,800 K, peaks near 0.5 µm in the visible band; surfaces near room temperature peak near 10 µm in the long-wave (thermal) infrared. This difference is the basis of the greenhouse effect in a glass or plastic glazed greenhouse or solar dryer: glazing transmits much of the short-wave solar radiation but is nearly opaque to the long-wave radiation emitted by the crop and floor. It also explains selective absorber coatings on solar collectors, which absorb strongly at short wavelengths while emitting weakly at long wavelengths.
Real surfaces emit less than a black body. Emissivity ε is the ratio of a surface's emission to that of a black body at the same temperature, so E = εσT⁴. When radiation falls on a surface, fractions are absorbed (absorptivity α), reflected (reflectivity ρ) and transmitted (transmissivity τ), with α + ρ + τ = 1; for an opaque body τ = 0. Kirchhoff's law states that at a given wavelength and temperature, emissivity equals absorptivity. A grey surface is one whose emissivity does not vary with wavelength, so its total emissivity equals its total absorptivity for any incident radiation; most hand calculations assume grey, diffuse surfaces. Polished metals have low emissivity; oxidised metals, paints, building materials, water, skin and vegetation are generally high. Values vary with finish and temperature, so they must be taken from tables for the actual surface. Low-emissivity foils in roofs and reflective films in insulated panels reduce radiant heat transfer; white exterior paint reflects much of the sunlight and so reduces solar heat gain on cold stores and animal houses.
Radiation leaving one surface reaches another only if the second is 'in view'. The view factor F12 is the fraction of the diffuse radiation leaving surface 1 that strikes surface 2 directly. It is purely geometric. Three rules let most view factors be found from a few chart values. Reciprocity: A1F12 = A2F21. Summation: for an enclosure, the view factors from any surface to all surfaces sum to one. A flat or convex surface cannot see itself, so F11 = 0; a concave surface can. For example, a small body inside a large enclosure has F12 = 1, and a disc of radius r covered by a hemisphere of the same radius has F(disc → hemisphere) = 1 and, by reciprocity, F(hemisphere → disc) = πr²/(2πr²) = 0.5, so half of what the hemisphere emits returns to itself. Net exchange between two black surfaces is Q12 = A1F12σ(T1⁴ − T2⁴). For grey surfaces the network method uses a surface resistance (1 − ε)/(εA) at each surface and a space resistance 1/(A1F12) between them. The results used most often are: two large parallel plates, q = σ(T1⁴ − T2⁴)/(1/ε1 + 1/ε2 − 1); a small grey body in a large enclosure, Q = ε1A1σ(T1⁴ − T2⁴); and long concentric cylinders or concentric spheres, Q = A1σ(T1⁴ − T2⁴)/(1/ε1 + (A1/A2)(1/ε2 − 1)). A radiation shield placed between two parallel plates adds two surface resistances; with N shields of the same emissivity as the plates, the exchange falls to 1/(N + 1) of the unshielded value. Where radiation and convection act together, a linearised radiation coefficient h_r = εσ(T1 + T2)(T1² + T2²) can be added to h.
A heat exchanger transfers heat between two fluids separated by a wall. In a double-pipe (tube-in-tube) exchanger one fluid flows in the inner tube and the other in the annulus, either in the same direction (parallel flow) or in opposite directions (counterflow). Shell-and-tube exchangers put many tubes inside a shell, with baffles to direct the shell fluid across the tubes; they are described by the number of shell passes and tube passes. Plate heat exchangers, the standard for milk and juice pasteurisation, use corrugated plates clamped together; they give high coefficients, compact size and easy cleaning, and allow regeneration, in which hot pasteurised milk preheats incoming raw milk. Cross-flow exchangers, such as car radiators and refrigeration coils, have the fluids moving at right angles; each fluid may be mixed or unmixed. The overall heat-transfer coefficient U combines all resistances between the two fluids. For a tube, 1/(U_o A_o) = 1/(h_i A_i) + R_fi/A_i + ln(r_o/r_i)/(2πkL) + R_fo/A_o + 1/(h_o A_o), where R_f are fouling resistances. For a thin wall, 1/U ≈ 1/h_i + t/k + 1/h_o. U is always less than the smaller of h_i and h_o, so improving the side with the lower coefficient gives the greatest gain.
Heat lost by the hot fluid equals heat gained by the cold fluid (neglecting losses): Q = m_h c_ph (T_h,in − T_h,out) = m_c c_pc (T_c,out − T_c,in). The product m c_p is the heat-capacity rate C, in W/K. Because the temperature difference between the fluids changes along the exchanger, the rate equation uses a mean: Q = U A ΔT_lm, where ΔT_lm = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2) is the log mean temperature difference. For parallel flow ΔT1 = T_h,in − T_c,in and ΔT2 = T_h,out − T_c,out; for counterflow ΔT1 = T_h,in − T_c,out and ΔT2 = T_h,out − T_c,in. The arithmetic mean always overestimates the log mean, and the error grows as the ratio ΔT1/ΔT2 grows. If ΔT1 = ΔT2, as in a counterflow exchanger with equal heat-capacity rates, the LMTD equals that constant difference. For the same inlet and outlet temperatures, counterflow gives a larger LMTD than parallel flow and so needs less area; counterflow also allows the cold fluid to leave hotter than the hot fluid leaves, which parallel flow can never do. When one fluid condenses or boils at constant temperature, the flow arrangement makes no difference to the LMTD. For multipass shell-and-tube and cross-flow exchangers the counterflow LMTD is multiplied by a correction factor F (less than or equal to 1) read from charts using two ratios, P and R; designs with F below about 0.75 are usually rejected. The LMTD method suits sizing problems, in which all four terminal temperatures are known or follow from the energy balance and the required area is sought.
When the outlet temperatures are unknown, as in rating an existing exchanger or checking how a pasteuriser performs at a new flow rate, the LMTD method needs trial and error. The effectiveness–NTU method avoids this. The maximum possible heat transfer is Q_max = C_min (T_h,in − T_c,in), because only the fluid with the smaller heat-capacity rate can undergo the full temperature difference; that would happen in an infinitely long counterflow exchanger. Effectiveness is ε = Q/Q_max, a number between 0 and 1. The number of transfer units, NTU = U A/C_min, measures the thermal size of the exchanger. The capacity ratio is C_r = C_min/C_max. For every flow arrangement, ε is a function of NTU and C_r only, so once ε is found, Q = ε C_min (T_h,in − T_c,in) and both outlet temperatures follow from the energy balances. Key relations are: parallel flow, ε = [1 − exp(−NTU(1 + C_r))]/(1 + C_r); counterflow, ε = [1 − exp(−NTU(1 − C_r))]/[1 − C_r exp(−NTU(1 − C_r))], which for C_r = 1 becomes ε = NTU/(1 + NTU); and for any arrangement with C_r = 0 (one fluid condensing or boiling, so its C is effectively infinite), ε = 1 − exp(−NTU).
A typical rating problem: water at a known flow enters a counterflow exchanger to cool milk at a known flow; U and A are known. Compute C_h and C_c, choose C_min, compute NTU and C_r, get ε from the relation, and then Q and the outlet temperatures. The ε–NTU curves teach three design lessons. First, effectiveness rises steeply at small NTU and then flattens, so beyond an NTU of about 3 to 5 a large increase in area buys a small gain in heat recovered. Second, for any NTU and C_r > 0 the counterflow arrangement gives the highest effectiveness, and parallel flow the lowest; for C_r = 1 parallel flow can never exceed ε = 0.5, however large the exchanger. Third, when C_r = 0 all arrangements give the same effectiveness. The regeneration section of a milk pasteuriser is a nearly balanced counterflow exchanger (raw and pasteurised milk have almost the same flow and specific heat), so its effectiveness is NTU/(1 + NTU); the dairy industry expresses it as regeneration efficiency (the temperature rise of raw milk in the regeneration section divided by the total rise needed to reach pasteurisation temperature), and high values save heating and cooling energy. The inverse problem, sizing, can also be done with ε–NTU: find the required ε from the desired duty, invert the relation for NTU, then A = NTU C_min/U.
Mass transfer is the movement of a chemical species caused by a difference in its concentration. Concentration can be expressed as mass concentration ρ_A (kg/m³), molar concentration C_A (kmol/m³), mass fraction, mole fraction or, for a gas, partial pressure p_A, since for an ideal gas C_A = p_A/(RT). Molecular diffusion obeys Fick's first law: the molar flux relative to the mixture's mean motion is J_A = −D_AB dC_A/dx, where D_AB is the binary diffusion coefficient (diffusivity) in m²/s. Orders of magnitude are important: in gases D is about 10⁻⁵ m²/s (for water vapour in air at room conditions about 2.5 × 10⁻⁵ m²/s); in liquids about 10⁻⁹ m²/s; in solids and in food matrices much smaller, often 10⁻¹⁰ to 10⁻¹² m²/s or lower. Gas diffusivity rises with temperature (as roughly T^1.5 to T^1.75 depending on the correlation) and falls inversely with total pressure. In liquids it rises with temperature and falls with viscosity. Fick's law is the exact analogue of Fourier's law, so plane-wall and cylinder results carry over: steady diffusion through a stationary plane layer is N_A = D_AB (C_A1 − C_A2)/L, and a resistance L/(D_AB A) can be defined.
In a gas the total flux of A includes bulk flow of the mixture as well as diffusion. Two limiting cases are standard. In equimolar counter-diffusion, for example in some distillation situations, every mole of A moving one way is matched by a mole of B moving the other way, so there is no net bulk flow and N_A = D_AB (p_A1 − p_A2)/(R T L). In diffusion of A through stagnant (non-diffusing) B, such as water evaporating from the bottom of a tube into air (the Stefan tube), the evaporating vapour induces a bulk flow towards the open end, which increases the flux: N_A = [D_AB P/(R T L)] ln[(P − p_A2)/(P − p_A1)], or equivalently N_A = D_AB P (p_A1 − p_A2)/(R T L p_B,lm), where p_B,lm is the log mean partial pressure of the stagnant gas. The factor P/p_B,lm is the bulk-flow correction; when A is dilute it is close to one and the simple Fick form is adequate. Transient diffusion obeys Fick's second law, ∂C/∂t = D ∂²C/∂x², the analogue of the transient conduction equation, so the Heisler-type and error-function solutions apply with α replaced by D and the Biot number replaced by the mass-transfer Biot number, h_m L/D.
Packaging films, fruit skins and waxed coatings limit the exchange of water vapour, oxygen and carbon dioxide. For a gas that dissolves in a film and then diffuses through it, the steady flux is proportional to the product of solubility and diffusivity, called permeability, and to the partial-pressure difference divided by thickness. Water-vapour transmission rate and oxygen transmission rate are measured by standard tests at stated temperature and humidity and are reported in units that vary between standards, so values from different sources must be converted carefully. Modified-atmosphere packaging relies on matching the film's permeability to the respiration rate of the produce so that oxygen and carbon dioxide settle at safe levels; the design is a mass-transfer balance between gas generation or consumption inside and diffusion through the film. At steady state the oxygen consumed by respiration equals the oxygen entering through the film, which fixes the internal oxygen partial pressure. Because respiration rate usually rises faster with temperature than film permeability does, a package designed for one storage temperature can become anaerobic if it is held warmer, which is a cold-chain as well as a packaging failure. Perforations change the balance sharply because diffusion through holes is gas-phase diffusion, orders of magnitude faster than diffusion through the polymer.
When a fluid flows past a surface where a species is being transferred, such as air over wet grain, water or a leaf, the rate is written like Newton's law of cooling: N_A = h_m A (C_A,s − C_A,∞), where h_m is the convective mass-transfer coefficient in m/s. The dimensionless groups mirror those of heat transfer. The Schmidt number, Sc = ν/D_AB, compares momentum and mass diffusivity and plays the role of Pr; for water vapour in air it is about 0.6. The Sherwood number, Sh = h_m L/D_AB, is the dimensionless mass-transfer coefficient and plays the role of Nu. The Lewis number, Le = α/D_AB = Sc/Pr, compares thermal and mass diffusivities; for water vapour in air it is close to one (about 0.85 or so). The mass-transfer Stanton number is St_m = h_m/V = Sh/(Re Sc). The coefficient can also be written on a partial-pressure basis or a humidity basis, with different units, so it is important to check which driving force a quoted coefficient belongs to before using it. For an ideal gas the conversion between a concentration-based and a partial-pressure-based coefficient uses C = p/(RT). Mass-transfer coefficients, like heat-transfer coefficients, are not properties: they depend on velocity, geometry and flow regime.
Because the governing equations are the same in form, any heat-transfer correlation Nu = C Re^m Pr^n becomes a mass-transfer correlation Sh = C Re^m Sc^n for the same geometry and flow regime, provided the transfer rate is low enough not to disturb the velocity field. For example, laminar flow over a flat plate gives Sh_avg = 0.664 Re^0.5 Sc^(1/3). The Chilton–Colburn analogy states j_H = St Pr^(2/3) = j_D = St_m Sc^(2/3) = C_f/2, valid roughly for 0.6 < Pr < 60 and 0.6 < Sc < 3,000. Dividing the heat and mass forms gives h/h_m = ρ c_p Le^(2/3). For air–water systems Le^(2/3) is close to one, so h/(h_m ρ c_p) ≈ 1: the Lewis relation. This is why the wet-bulb temperature of air–water vapour mixtures is very close to the adiabatic saturation temperature, which is what makes the psychrometric chart practical. The analogy is widely used in practice: if the convective heat-transfer coefficient over a product in a dryer is known, the mass-transfer coefficient follows, and vice versa. The analogy fails when properties change strongly, when there is chemical reaction at the surface, or when the transfer rate is so high that it 'blows' the boundary layer.
A wet surface exposed to unsaturated air loses water by convective mass transfer and must receive the latent heat of vaporisation. At steady state the heat received by convection (and radiation) equals the latent heat carried away: h (T_∞ − T_s) = h_m h_fg (ρ_v,s − ρ_v,∞). The surface settles at the wet-bulb temperature, below the air temperature. This balance governs the wet-bulb thermometer, pad-and-fan evaporative coolers for greenhouses and poultry houses, the evaporative (zero-energy) cool chambers used for produce storage in hot dry climates, and the cooling of cooling-tower water. Evaporative cooling works best in dry air, because the air's capacity to take up water depends on its vapour-pressure deficit; in humid weather the wet-bulb depression is small and so is the cooling effect. The lowest temperature an ideal direct evaporative cooler can reach is the wet-bulb temperature of the incoming air; real pads reach a fraction of the wet-bulb depression, expressed as saturation efficiency. Evaporative cooling also raises the humidity of the air, which is beneficial for most fresh produce, which loses less water in humid air, but can be harmful for animals under heat stress when the air is already humid.
Drying removes water by evaporation, so heat must reach the evaporating water and the vapour must leave the product. Moisture content is expressed on a wet basis, M_wb = water/(water + dry matter), or on a dry basis, M_db = water/dry matter; they are related by M_db = M_wb/(1 − M_wb). Dry basis is used in drying calculations because the dry-matter mass stays constant. A hygroscopic product held in air of fixed temperature and relative humidity eventually reaches its equilibrium moisture content, which is set by the product's sorption isotherm; values depend on the product, variety and whether it is gaining or losing moisture (hysteresis). A plot of drying rate against moisture content has, in the classic case, a short settling period, a constant-rate period and one or more falling-rate periods. The moisture content at which the rate starts to fall is the critical moisture content. It is not a fixed property of a material: it rises with thicker layers and faster drying, because the surface dries out earlier when internal transport cannot keep up. Water that is bound in the product's structure has a lower vapour pressure than free water at the same temperature, which is why hygroscopic products need more energy per kilogram of water removed near the end of drying.
During the constant-rate period the surface stays wet, and water evaporates as from a free water surface. The surface is at the wet-bulb temperature of the air, and the rate is controlled entirely by external conditions: air temperature, humidity and velocity. The drying rate per unit area is R_c = h (T_a − T_wb)/h_fg = k_y (Y_s − Y_a), the heat and mass forms of the same balance. The time to dry from X_1 to X_c in this period is t = (W_s/A)(X_1 − X_c)/R_c, where W_s is dry-solid mass and X is dry-basis moisture. Raising air temperature or velocity, or lowering its humidity, raises the constant rate. Many cereal grains and most dried plant materials show little or no constant-rate period at harvest moisture; their drying is mostly falling-rate. The constant-rate period is most visible with very wet materials such as washed vegetables, filter cakes, pulps and non-hygroscopic granular solids. Because the surface stays at the wet-bulb temperature, even hot air does not overheat the product during this period, which is why higher air temperatures can be used safely early in a drying run and reduced later.
Once the surface is no longer continuously wet, water must move from the interior to the surface by liquid diffusion, capillary flow or vapour diffusion, and the rate falls. The product surface warms towards the air temperature, which is why product temperature must be watched late in a drying run. If the rate falls linearly with moisture, the time from X_c to X is t = (W_s/A)[(X_c − X_e)/R_c] ln[(X_c − X_e)/(X − X_e)]. If internal diffusion controls, Fick's second law applies with an effective diffusivity D_eff, which lumps all internal transport mechanisms. The moisture ratio MR = (M − M_e)/(M_0 − M_e) for long times is approximately (8/π²) exp(−π² D_eff t/(4L²)) for a slab of half-thickness L drying from both faces, and (6/π²) exp(−π² D_eff t/r²) for a sphere of radius r. These show that drying time scales with the square of thickness: halving slice thickness cuts the time to about a quarter. D_eff is found by plotting ln MR against time and taking the slope; its temperature dependence is usually described with an Arrhenius equation. In grain drying, empirical thin-layer models are used: the Lewis (Newton) model MR = exp(−kt), the Page model MR = exp(−k t^n) and the Henderson–Pabis model MR = a exp(−kt), with constants fitted to experimental data. Deep-bed drying is then modelled as a series of thin layers, and a drying front moves through the bed, with the lowest layers overdried and the top layers still wet; mixing or reversing the airflow reduces this gradient.
The energy needed to evaporate water is roughly the latent heat of vaporisation, about 2.26 MJ/kg at 100 °C and about 2.5 MJ/kg near 0 °C, plus extra energy to remove bound water from hygroscopic material, plus heating of the product and losses in the exhaust air. Dryer thermal efficiency is therefore well below one, and recirculating part of the exhaust air or using heat recovery improves it. High air temperature speeds drying but can cause case hardening (a dry crust that slows further drying), stress cracks in rice and maize, browning and loss of heat-sensitive nutrients and germination. Safe drying temperatures depend on whether grain is for seed, milling or feed; national guidelines and the product's own data should be followed. Tempering, a rest period between drying passes, lets moisture inside kernels redistribute so that the next pass removes water faster and with less cracking; this is standard practice in paddy drying. Solar dryers use the same principles at lower temperature: the collector heats air by absorbing short-wave radiation, and the chimney or fan moves it through the product.
A student must be able to set up a resistance network for any wall or pipe, choose the right characteristic length for each dimensionless number, and decide quickly whether a problem is steady or transient, lumped or not, LMTD or ε–NTU. Practise with units written at every step, because most lost marks are unit slips: kJ/h against W, diameter against radius, Celsius against kelvin in radiation. Build a one-page sheet of the key formulas in this manual and test it against worked problems from a standard textbook until each setup takes under a minute. Link every formula to a farm or food example — a cold-store wall, a milk cooler, a grain dryer — so the numbers have meaning.
An engineer five years into practice uses this subject to size and check equipment: calculating insulation thickness and heat gain for a cold room or ripening chamber, checking whether a plate heat exchanger can pasteurise more milk after a capacity increase (an ε–NTU rating), estimating fouling allowance and cleaning intervals, sizing the air heater and fan for a batch grain dryer, and predicting the pre-cooling time of produce in forced-air or hydro-coolers from the half-cooling time. The practitioner reads manufacturer data critically, knows that U quoted for clean surfaces falls in service, and verifies performance by measuring temperatures and flows on site and doing an energy balance.
A senior engineer with 25 years of experience uses heat and mass transfer for system-level decisions: choosing between insulated panel types for a cold chain on life-cycle energy cost, specifying regeneration efficiency for a new dairy line, setting drying-air temperatures that balance throughput against cracking and germination loss, deciding where heat recovery pays, and diagnosing field failures such as condensation inside insulation, burnout of heater elements, frost build-up on evaporators or uneven drying in deep beds. They also write specifications and tender evaluations that state performance in testable heat-transfer terms, and they train younger engineers to check designs with simple resistance and energy-balance estimates before trusting software.
A researcher at doctoral level with long field experience works on the parts of the subject that textbooks treat as solved but that fail for biological materials: shrinkage and moving boundaries during drying, variable effective diffusivity, coupled heat, moisture and respiration in stored produce, porous-media models for grain bins, microwave and radio-frequency heating, and computational fluid dynamics of cold stores and dryers. They design experiments to measure properties (thermal conductivity, diffusivity, sorption isotherms) under controlled conditions, validate models against data with honest uncertainty, and judge when an empirical thin-layer model is adequate and when a mechanistic model is needed for scale-up or control.
This manual does not reproduce Heisler or view-factor charts, heat-exchanger correction-factor charts or property tables, because figures and tables are not yet supported; students must use a textbook for them. It does not cover numerical (finite-difference) methods for conduction in depth, two-phase flow boiling in tubes, gas radiation (participating media), mass transfer with chemical reaction, or the design of cooling towers and evaporators beyond the underlying principles. Property values quoted are orders of magnitude only; real designs need measured or tabulated values for the specific material and temperature. Microwave, radio-frequency and ohmic heating are only mentioned.
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. Every numerical answer was computed by script; property values are given in each stem and are illustrative, not design data.
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