Heat, Mass Transfer and Food Preservation

Thermal process lethality (F0 and D-z values)

Thermal process lethality expresses the microbial destruction achieved by a heat treatment as an equivalent time at a fixed reference temperature. In canning, the reference temperature is normally 121.1 °C, and F_0 is the number of minutes at 121.1 °C that would produce the same kill as the actual heating cycle at the slowest-heating point of the container. For a time-temperature history, lethality is accumulated as F = \sum \Delta t \cdot 10^{(T - T_{ref})/z}, so each time interval contributes according to how far its temperature lies above or below the reference.

• D value — the time in minutes for a one-log (90 %) reduction of a population at a fixed temperature. For constant-temperature heating, survivors follow N = N_0\,10^{-t/D}, so the heating time is t = D\log_{10}(N_0/N).
• Lag time — in the microbial death equation \log(N_0/N) = 1 + (t - t_l)/D, setting t = t_l gives N = 0.1\,N_0. The lag time is therefore the time for a tenfold reduction, that is a 90 % kill, and not a 10 % reduction.
• z value — the temperature change that alters D by a factor of ten. When two treatments give equal lethality, t_1\,10^{(T_1 - T_{ref})/z} = t_2\,10^{(T_2 - T_{ref})/z}, so z = (T_1 - T_2)/\log_{10}(t_2/t_1). A short high-temperature treatment and a longer low-temperature treatment with the same lethality illustrate this relation directly.

For inoculated packs, the same lethality means the same F-value is delivered to each organism, so the required reduction for each organism is \log_{10}(N_{0}/N_t) = F/D. If spoilage probability is P for Poisson-distributed survivors, the expected survivors per container are N_t = -\ln(1-P). The organism with the larger D value needs the larger initial load to produce the same survivor count.

Psychrometric humidity and saturation

The humidity ratio of moist air is H = 0.622\,p_v/(P - p_v), where p_v is the partial pressure of water vapour, P is the total pressure, and 0.622 = M_w/M_a \approx 18.02/28.97. Relative humidity is the ratio of the actual vapour pressure to the saturation pressure at the same dry-bulb temperature, \phi = p_v/p_{sat}(T). Relative humidity does not depend on total pressure and is calculated directly from partial pressures.

The percentage absolute humidity is defined as 100\,H/H_s, where H_s is the humidity ratio at saturation. Since H/H_s = (p_v/p_s)(P - p_s)/(P - p_v), the two percentages coincide when the second factor equals unity. This occurs at saturation, where p_v = p_s, and in very dry air, where both pressures are negligible compared with P.

When two air streams are mixed, the humidity ratio of the mixture is the average of the individual humidity ratios weighted by their dry-air mass flow rates, W_m = (\dot m_1 W_1 + \dot m_2 W_2)/(\dot m_1 + \dot m_2). The dry-air mass flow is the volumetric flow divided by the specific volume of moist air, v = (R_a T/P)(1 + 1.6078\,W).

Psychrometric processes change dry-bulb temperature (DBT) and dew-point temperature (DPT) in characteristic ways:

• Sensible cooling — DBT falls while the moisture content, and hence the DPT, stays constant.
• Cooling and dehumidification — both DBT and DPT fall as water condenses.
• Chemical dehumidification — a desiccant removes vapour and releases latent heat, so DBT rises while DPT falls.
• Heating and humidification — both DBT and DPT rise when steam is added to heated air.

For example, if air has p_v = 2 kPa and p_s = 4 kPa at the same temperature, then \phi = 50\% and H = 0.622 \times 2/(P - 2). Converting between these quantities is the core skill tested in psychrometric problems.

Heat exchanger types

Heat exchangers are classified by construction and flow arrangement, and each type suits different duties:

• Pipe-in-pipe (double-pipe) exchanger — has a small heat transfer area, suited to small duties.
• Shell and tube exchanger — handles large flow rates because of its large flow area and surface.
• 1-2 shell and tube exchanger — one shell pass and two tube passes allow the two fluids to experience both co-current and counter-current regions, which gives a correction factor below one.
• Cross-flow exchanger — the two fluids flow at right angles, and it is commonly used for cooling air.

The log mean temperature difference (LMTD) method gives the heat transfer rate as q = UA\,F\,\Delta T_{lm}. The correction factor F accounts for departures from pure counter-flow or parallel-flow, and it equals 1 for a single shell pass with a single tube pass. For a counter-flow exchanger, \Delta T_{lm} = (\Delta T_1 - \Delta T_2)/\ln(\Delta T_1/\Delta T_2), where \Delta T_1 and \Delta T_2 are the terminal temperature differences.

The heat duty can be found from the hot-stream energy balance q = \dot m_h c_{p,h}(T_{h,in} - T_{h,out}). The outlet temperature of the cold stream then follows from q = \dot m_c c_{p,c}(T_{c,out} - T_{c,in}), and the area is A = q/(U\,\Delta T_{lm}). For example, a counter-flow pasteurizer with a 2 kW/m^2K coefficient needs an area set by the duty and the LMTD, not by the inlet temperatures alone.

In the effectiveness-NTU approach, NTU = UA/C_{min}, and the heat transfer is q = \varepsilon\,C_{min}(T_{h,in} - T_{c,in}). The minimum heat capacity rate C_{min} is the smaller of the two stream capacity rates, C = \dot m c_p, and it determines the maximum possible heat transfer. Knowing \varepsilon and NTU, the outlet temperature of the hot stream follows from its own capacity rate.

Membrane mass transfer

Mass transfer through a membrane is often modelled as a series of resistances: the boundary-layer film on the feed side, the membrane itself, and the boundary-layer film on the permeate side. The overall solute flux is N = C_0/(1/k_1 + 1/P + 1/k_2), where k_1 and k_2 are the film mass transfer coefficients, P is the membrane permeability, and C_0 is the feed concentration difference. The membrane thickness does not appear separately when the permeability is supplied, because P already combines the diffusivity and the thickness of the membrane.

The removal rate of solute is the flux multiplied by the membrane area, \dot m = N\,A. If the permeate side is at zero concentration, the driving force equals the feed concentration, and the sum of resistances determines how much solute passes per unit time.

For a gas that permeates a multilayer package, the layers act in series and the permeation rate is \dot V = \Delta p\,A/\sum(L_i/P_i). Each layer contributes L_i/P_i to the total resistance, so the most resistive layer controls the barrier. The shelf life against a fixed permeated quantity is the allowable volume divided by the permeation rate.

Heat transfer through slab with convective surfaces

A flat slab exposed to a fluid on each face is treated as a series thermal circuit. The resistances per unit area are 1/h_1 for the hot-side film, L/k for conduction through the slab, and 1/h_2 for the cold-side film. The total resistance is R = 1/h_1 + L/k + 1/h_2, and the steady heat flux is q = \Delta T_{overall}/R.

If the heat flow rate and area are known, the overall temperature difference is \Delta T = (Q/A)R. A film resistance can be as important as the conduction resistance when the film coefficient is low, so both convective resistances must be included.

For a composite wall made of several layers, each layer adds its own conduction resistance L_i/k_i in series. The heat flux is the total temperature difference divided by the sum of all layer resistances, and the layer with the lowest conductivity, such as a cork insulation, usually dominates the total resistance.

Freezing heat load and latent heat

The heat removed during freezing of a food is the sum of three stages: sensible cooling above the freezing point, latent heat removal at the freezing point, and sensible cooling below it. The total load is Q = m c_{p,u}(T_i - T_f) + m\,h_{sf} + m c_{p,f}(T_f - T_{final}), where h_{sf} is the latent heat of fusion and c_{p,u} and c_{p,f} are the specific heats above and below freezing.

A common error is to omit the latent heat, which is usually the largest term because the freezing point marks a large release of energy as water solidifies. Specific heat values differ on the two sides of the freezing point because the frozen fraction changes the composition of the product.

In scraped-surface freezers, ice builds up on the cooled wall at a rate given by q = U\,\Delta T. The mass of ice per unit area for a layer of thickness \delta is \rho\,\delta, and the energy needed to form it is \rho\,\delta\,h_{sf}. Dividing this energy by the heat flux gives the time to form the layer. With n blades passing a given point, the scraping interval is 60/(nN) s for a speed of N rpm, and the ice must be removed before the layer reaches its limit thickness.

Dimensionless numbers in forced convection

Dimensionless numbers group fluid and transport properties so that convective correlations can be generalised. The Prandtl number is Pr = \nu/\alpha = c_p\mu/k, the ratio of momentum diffusivity to thermal diffusivity. It combines the three fluid properties specific heat, dynamic viscosity and thermal conductivity in one group.

The Schmidt number Sc = \nu/D is the mass-transfer analogue of the Prandtl number, with D the mass diffusivity. Because both groups have momentum diffusivity in the numerator, they play analogous roles in heat and mass transfer correlations, with Pr and Sc replacing each other.

The Lewis number is Le = \alpha/D = Sc/Pr. It is a ratio, not a product, and it indicates whether heat or mass diffusion dominates in a process. A small Prandtl number indicates that thermal diffusivity is large compared with momentum diffusivity, so heat spreads faster than momentum.

Other groups in convection include the Reynolds number, which compares inertial and viscous forces, and the Froude number, which compares inertial and gravitational forces. The Biot number compares internal conduction with surface convection, and it contains conductivity but not specific heat or viscosity.

Critical radius of insulation

Adding insulation to a cylinder increases conduction resistance but reduces the outer convective resistance because the outer surface area grows. The critical radius is the outer radius at which the heat loss is a maximum. For a cylinder, r_{cr} = k/h_o, obtained by setting the derivative of the total resistance with respect to the outer radius to zero.