Food Engineering Operations

Drying and psychrometric calculations

Psychrometry describes mixtures of dry air and water vapour through three linked quantities: the humidity ratio, the relative humidity and the enthalpy. The humidity ratio is the mass of water vapour per unit mass of dry air, \omega = 0.622\,p_v/(P - p_v), where p_v is the partial pressure of vapour, P is the total pressure, and 0.622 = M_w/M_a \approx 18.02/28.97. Relative humidity is \phi = p_v/p_{sat}(T) at the same dry-bulb temperature, so the vapour pressure follows directly from \phi and the saturation pressure is taken from steam tables. The enthalpy per kg of dry air is h = c_{pa}T + \omega(h_{fg,0} + c_{pv}T), referred to liquid water at 0 °C, so the liquid specific heat does not appear.

Moisture in drying is reported on two bases. The dry basis, X_{db} = m_w/m_s, can exceed 100%, whereas the wet basis, X_{wb} = m_w/(m_w + m_s), is always below 100%. They convert as X_{db} = X_{wb}/(1 - X_{wb}), so a value quoted on one basis must never be inserted into a formula written for the other. Because dry solids are not lost in drying, a solids balance gives the product mass, and the water removed equals feed minus product. For example, 10 kg of feed at 20% wet basis holds 8 kg of dry solid; drying to 10% dry basis leaves 0.8 kg of water, so 1.2 kg of water is removed.

• dry basis — moisture relative to the mass of bone-dry solid. e.g. 1 kg water in 4 kg solid is 25% dry basis.
• wet basis — moisture relative to the total wet mass. e.g. the same sample is 20% wet basis.
• relative humidity — vapour pressure as a fraction of saturation pressure at the same temperature.

Centrifugal separation

A centrifuge replaces gravity with a radial acceleration, and the ratio of that acceleration to gravity is the relative centrifugal force, \text{RCF} = \omega^2 r/g. With speed in rpm, \omega = 2\pi N/60, so the g-force grows with the square of speed and linearly with radius. For example, a bowl of 0.05 m radius at 3000 rpm gives about 500 g, which is why high-speed centrifuges separate fine emulsions that would never settle under gravity.

In liquid–liquid separation in a tubular bowl, the light and heavy phases leave through separate outlets, and between them lies a neutral zone (interface) where the phase pressures balance. Its radius satisfies r_n^2 = (\rho_H r_H^2 - \rho_L r_L^2)/(\rho_H - \rho_L), where \rho_H and r_H refer to the heavier phase and its outlet, and \rho_L and r_L to the lighter phase and its outlet. The neutral zone always lies between the two outlet radii, and a larger density difference moves it toward the lighter-phase outlet.

The capacity of a centrifuge is described by the equivalent clarification area \Sigma, defined so that \Sigma = Q/v_t, where Q is the volumetric feed rate and v_t is the settling velocity of the critical particle. On a gravity-settling basis, Stokes' law gives v_t = d^2(\rho_p - \rho_f)g/(18\mu), so the required area rises with throughput and falls with the square of particle diameter.

• RCF — the g-force generated by rotation; it is not the same as the rpm value.
• neutral zone — the radius at which the two liquid phases balance.
• clarification area — the equivalent gravity-settling area that gives the same clarification.

Heat transfer in pipe flow

Heat transfer in a pipe carrying a process fluid such as milk is analysed first with an energy balance and then with a flow regime check. The heat duty is Q = \dot m\, c_p\,\Delta T, so the mass flow rate follows from a measured temperature drop and a known heat loss. The Reynolds number for a circular pipe is Re = 4\dot m/(\pi D\mu), which is equivalent to \rho V D/\mu with mean velocity V. Flow is laminar below about 2100 and turbulent above about 4000. Re depends on the mass flow and not on the pipe length, so the length is not needed for this step.

For a tube wall, the overall heat transfer coefficient referred to the inside area follows from resistances in series: 1/U_i = 1/h_i + (r_i/k)\ln(r_o/r_i) + (r_i/r_o)(1/h_o). The conduction term uses the logarithmic geometry of a cylinder, and the outer film resistance is scaled by r_i/r_o when it is referred to the inside area. Rearranging this equation gives the unknown outside film coefficient when the others are known.

In a counterflow double-pipe exchanger, the temperature differences at the two ends are \Delta T_1 and \Delta T_2, and the log mean temperature difference is \text{LMTD} = (\Delta T_1 - \Delta T_2)/\ln(\Delta T_1/\Delta T_2). In counterflow, the end pairs are the hot inlet with the cold outlet, and the hot outlet with the cold inlet. Counterflow gives a larger LMTD than parallel flow for the same terminal temperatures, which is why it is preferred.

• LMTD — the effective driving force for a heat exchanger with varying temperatures.
• overall coefficient U — combines film and wall resistances; it must be referred to a stated area.

Food processing unit operations

Each food unit operation is recognised by the change it produces in the material and by its typical product. Matching a process to its application is a standard recall topic.

• Hydrogenation — catalytic addition of hydrogen to unsaturated oil, which raises the melting point and gives a semi-solid fat used as shortening.
• Blanching — a brief hot-water or steam treatment that inactivates enzymes such as peroxidase and lipoxygenase, fixes colour and reduces microbial load before freezing or drying.
• Leaching — solid–liquid extraction in which a solvent dissolves a soluble component, such as a colouring matter, out of a solid matrix.
• Winterization — slow chilling of oil so that high-melting waxes and stearin crystallise and are removed by filtration or centrifugation, which prevents clouding at low temperature.
• Fermentation — microbial conversion of sugars that leavens or acidifies a product; idli batter is leavened by lactic acid bacteria and yeasts.
• Parboiling — soaking, steaming and drying of paddy so that the starch gelatinises inside the grain before milling.
• Carbonation — dissolving carbon dioxide under pressure in a beverage to give effervescence.
• Baking — dry heat in an oven that sets the structure and browns the product, as in biscuits and cookies.

Fluid flow dimensionless numbers

The Reynolds number is the ratio of inertial forces to viscous forces, and it decides whether pipe flow is laminar or turbulent. It is written as Re = \rho V D/\mu, where \rho is density, V is mean velocity, D is the internal diameter and \mu is dynamic viscosity. Dividing \mu by \rho gives the kinematic viscosity \nu, so the same number can also be written as Re = VD/\nu. Any expression that is correct must have the viscosity in the denominator; placing viscosity in the numerator gives a reciprocal, which is dimensionally wrong.

For water in a pipe, the Reynolds number is evaluated with consistent SI units. For example, water with \rho = 1000 kg/m^3 and \mu = 10^{-3} Pa·s at a mean velocity of 1 m/s in a 0.02 m pipe gives Re = 20\,000. The mean velocity is obtained from the volumetric flow rate divided by the cross-sectional area \pi D^2/4, and this step must be done before the Reynolds number is calculated.

• kinematic viscosity — \nu = \mu/\rho, with units of m^2/s.
• volumetric flow to velocity — V = Q/A, with A = \pi D^2/4 for a circular pipe.

Evaporator mass and energy balance

The design of a single-effect evaporator rests on a solids balance and an energy balance. Dissolved solids are conserved, so F x_F = P x_P, where F and P are feed and product mass flows and x is the solids fraction. The water evaporated is V = F - P. Concentrating juice from 10% to 40% solids, for example, gives a product of one quarter the feed mass, and the water evaporated is the remaining three quarters.

The energy balance states that the steam heat supplied equals the latent heat carried away by the vapour, plus the enthalpy carried by the concentrate, minus the enthalpy of the feed. Written with enthalpies referred to 0 °C, Q = V H_v + P h_c - F h_f. When the product leaves at its boiling temperature, this can be rewritten as Q = V(H_v - h_c) + F(h_c - h_f). Operating under vacuum lowers the boiling point, which allows heat-sensitive juices to be concentrated at moderate temperature.

When concentration is specified on a wet basis, the dry solids are the conserved quantity. The product weight is the dry solid mass divided by (1 - X_{wb}) of the product, and the water removed is the difference between the feed and product weights.

Refrigeration heat load calculation

The capacity of a cold storage plant equals the heat load it must remove per unit time, adjusted for the plant efficiency. The load has several parts. The sensible heat removed from the product is Q_s = m\,c_p\,\Delta T. For respiring produce such as potatoes, the heat of respiration over the storage period adds a further term m\,q_r, where q_r is the heat released per unit mass. Dividing the total energy by the time in seconds gives the average load in kW.

The refrigeration unit is expressed in tons of refrigeration (TR), where 1 TR = 3.517 kW. The required capacity is the load divided by the stated plant efficiency, so a lower efficiency means a larger installed capacity. Converting the result to TR is the final step.

Heat gain through a wall is estimated by steady conduction through each layer in series. The heat flux is q = \Delta T/\sum_i (L_i/k_i), where L_i and k_i are the thickness and conductivity of each layer. Insulation with a low conductivity contributes the largest resistance, so it dominates the total heat flux.

Size reduction equipment

Size reduction machines are classified by the mechanism that breaks the particles, and the mechanism determines which equipment suits a given material.

• Hammer mill — breaks particles by impact from rapidly rotating hammers; used for dry, brittle materials such as grains and spices.
• Burr mill — reduces particles by attrition, that is, rubbing between two grinding surfaces.
• Crushing rolls — reduce particles by compression between two counter-rotating rolls.
• Rotary knife cutter — cuts material by shearing between knives, giving a defined particle shape.

Size reduction also depends on the energy–size relation. Kick's law assumes that energy is proportional to the logarithm of the size ratio, and Rittinger's law assumes it is proportional to the new surface created. Bond's law, treated in the next section, gives an intermediate description for typical grinding.

Moisture sorption isotherms

A moisture sorption isotherm plots equilibrium moisture content against water activity at a fixed temperature. Water activity a_w is the vapour pressure of the food divided by that of pure water at the same temperature, and it governs microbial growth and chemical stability. Isotherms are often fitted with the Guggenheim–Anderson–de Boer (GAB) model, which describes multilayer sorption over a wide range of water activity.