Mass, Momentum and Heat Transfer in Food Engineering

Reynolds number in pipe flow

The Reynolds number is a dimensionless group that compares inertial forces with viscous forces in a flowing fluid. For flow in a uniform circular pipe, it is defined with the mean velocity and the internal diameter as Re = \frac{\rho V D}{\mu}, where \rho is density, V is the mean velocity, D is the internal diameter and \mu is the dynamic viscosity. Because kinematic viscosity is \nu = \mu/\rho, the same number can also be written as Re = \frac{VD}{\nu}. Students often confuse the Reynolds number with its reciprocal, so any expression that places viscosity in the numerator is not the Reynolds number. When the flow rate Q is given in volume per time, the mean velocity is V = Q/(\pi D^2/4), and all quantities must be converted to SI units before substitution. In pipe flow, the regime is generally laminar for Re below about 2300 and turbulent at higher values. Example: a fluid with kinematic viscosity 10^{-6} m²/s moving at a mean velocity of 0.5 m/s in a 0.04 m pipe has Re = (0.5)(0.04)/10^{-6} = 20000, which is turbulent.

Moisture content conversion

Moisture content of a food is reported on either a wet basis or a dry basis. On a wet basis, the moisture content is the mass of water divided by the total mass of the material, M_w = \frac{m_w}{m_w + m_d}, where m_d is the bone-dry solid mass. On a dry basis, it is the mass of water divided by the dry solid mass, M_d = \frac{m_w}{m_d}. The two are related by M_d = \frac{M_w}{1 - M_w}, so a wet-basis value is always smaller than the dry-basis value at the same moisture level. The key principle in all mass problems is that the dry matter is conserved during drying, rehydration and most processing steps, while the water content changes. Therefore, the dry matter at one moisture level equals m_{total}(1 - M_w) at that level, and this quantity can be equated before and after processing. The coefficient of rehydration compares the mass of the rehydrated product with the mass of the fresh material carrying the same dry matter. Example: 50 kg of a product at 20% wet basis contains 40 kg of dry matter; if this is dried to 10% wet basis, the total mass becomes 40/0.9 = 44.4 kg.

Fluid flow and heat transfer equations

Dimensionless groups in transport phenomena are ratios of physical properties and characteristic lengths, and they must be written with the correct property in each place. The Reynolds number is Re = \frac{\rho V L}{\mu}, and the Schmidt number is Sc = \frac{\nu}{D}, the ratio of momentum diffusivity to mass diffusivity. The Nusselt number is Nu = \frac{hL}{k_f}, which uses the thermal conductivity of the fluid k_f, whereas the Biot number is Bi = \frac{hL}{k_s}, which uses the thermal conductivity of the solid k_s. Both have the form hL/k, so the conductivity subscript distinguishes them. Fourier's law for one-dimensional steady conduction is q'' = -k\frac{dT}{dx}. For a flat plate of thickness L with a fixed temperature difference \Delta T, the gradient magnitude is \Delta T/L and the heat flux is q'' = k\Delta T/L. Thus the flux is directly proportional to the conductivity, and it falls as the thickness increases. The heat transfer area does not change the gradient; it only enters the total heat rate Q = kA\Delta T/L.

Supercritical fluid extraction

Supercritical fluid extraction (SFE) uses a fluid held above both its critical temperature and critical pressure. In this state the fluid has gas-like transport properties and liquid-like solvent power, and the solvent strength can be tuned by changing pressure and temperature. Carbon dioxide is the most common supercritical medium for spices, herbs and tea because its critical point, about 31.1 °C and 7.38 MPa, is mild, and CO₂ is non-toxic, non-flammable and inexpensive. Pure CO₂ is non-polar, so a small amount of a polar co-solvent such as ethanol is often added to extract polar constituents such as phenolic compounds. Hot pressurised water near or above its critical region is also used as a green extraction medium for plant material. Conventional organic solvents such as dichloromethane are liquid solvents and are not supercritical media; they are excluded from the SFE definition. Example: SFE with CO₂ and 5% ethanol co-solvent is used to recover polar antioxidants from dried herbs at moderate temperatures.

Size reduction equipment

Size reduction machines are classified by the mechanism that breaks the feed particles, and the mechanism must be matched to the equipment. Compression breaks material between two surfaces that squeeze it, as in crushing rolls and jaw crushers. Impact breaks material by striking it at high speed, as in hammer mills and impact breakers. Attrition breaks material by rubbing it between two grinding surfaces, as in burr mills and disc mills. Cutting shears material with blades, as in rotary knife cutters. Hammer mills are therefore impact machines, burr mills are attrition machines, crushing rolls are compression machines, and rotary knife cutters are cutting machines. Recognising the mechanism is the main step in answering matching questions, because the physical name of the machine often reveals its action directly.

Heat loss in milk pipelines

When milk flows through an insulated or uninsulated pipe, the heat lost to the surroundings reduces the milk temperature along the line. The heat balance for a steady flow is Q = \dot m c_p \Delta T, where \dot m is the mass flow rate, c_p is the specific heat capacity and \Delta T is the temperature drop. If the heat loss and temperature change are known, this relation gives the mass flow rate directly. For a circular pipe, the mean velocity satisfies \dot m = \rho V A, so \rho V d/\mu = 4\dot m/(\pi d\mu). This means the Reynolds number can be found from the mass flow rate without needing density separately. Once the Reynolds number is computed, the flow regime is judged against the laminar limit of about 2300. Milk is usually handled in turbulent flow in processing lines, but low-flow sections and small tubes can be laminar, and the regime affects the heat transfer coefficient.