Question 1 of 9: Thermal Properties in Engineering Design, the Cp–Enthalpy Relationship, Thermal Conductivity of Foods, and the Heat Transfer Coefficient

Question 1: Thermal Properties in Engineering Design, the Cp–Enthalpy Relationship, Thermal Conductivity of Foods, and the Heat Transfer Coefficient (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part (a) — Why thermal properties matter in design. Every unit operation that heats, cools, freezes, dries or thermally processes a food — pasteurizers, retorts, blast freezers, ovens, evaporators — is sized by solving the heat (and often mass) balance for that product, and every term in that balance is a thermal property: thermal conductivity k and thermal diffusivity α set how fast heat penetrates to the product's centre, specific heat c_p sets how much energy must be added or removed per degree of temperature change, and latent heat sets the energy cost of a phase change (freezing, boiling). Sizing a retort's process time (and hence the lethality delivered to the coldest point of a can) without an accurate k/α either under-processes (a food-safety failure) or over-processes (a quality and energy-cost failure); sizing a freezer's refrigeration duty without accurate c_p and latent heat mis-sizes the compressor. Thermal properties are also composition- and temperature-dependent (they change with moisture content, fat content and the fraction of the water that is frozen), so a design that uses a generic literature value rather than a property measured (or predicted from composition) for the actual product risks a systematic under- or over-design.

Part (b) — Cp and enthalpy. Specific heat c_p is, conceptually, how much sensible heat a unit mass of the food stores per degree of temperature rise at constant pressure; enthalpy H is the total heat content of that mass relative to some reference state. Thermodynamically the two are directly linked as a derivative and its integral: $$\begin{aligned} c_p &= \left(\frac{\partial H}{\partial T}\right)_P, \\ \Delta H &= \int_{T_1}^{T_2} c_p\, dT. \end{aligned}$$ For a food that is not changing phase over the interval, c_p is nearly constant and this collapses to the familiar ΔH = m c_pΔT used in Question 1's calorimetry-style balances elsewhere in this subject. Where a phase change (ice melting, fat crystals melting) occurs within the temperature range, c_p diverges sharply around the transition (an "apparent specific heat" peak) because a large amount of latent enthalpy is absorbed over a narrow temperature band with almost no temperature rise; the enthalpy–temperature curve is therefore the more fundamental and better-behaved property for a freezing/thawing calculation, and reported "apparent c_p" data for a food's freezing range are really the local slope of that enthalpy curve.

Part (c) — Dominant component for thermal conductivity. Water content is, for almost every food, the single most important compositional determinant of k. Water's own thermal conductivity (≈0.6 W/(m·K) at room temperature, and roughly four times higher again once frozen to ice, ≈2.2 W/(m·K)) is far higher than that of the other major food components — protein, carbohydrate, fat and air all sit in the 0.15–0.35 W/(m·K) range, and entrapped air (in a porous or aerated product) is lower still, ≈0.026 W/(m·K). Because water is both the highest-conductivity component and typically the majority component by mass in most fresh foods, composition-based predictive models (e.g. Choi & Okos' parallel/series mixture models) show k rising almost linearly with moisture content, and freezing a food's water raises its bulk k substantially — which is exactly why frozen-food heat-transfer calculations cannot reuse the unfrozen product's k.

Part (d) — Heat transfer coefficient. Conceptually, the (convective) heat transfer coefficient h lumps everything about a moving fluid's boundary layer (velocity, turbulence, fluid properties) into a single number that converts a surface-to-fluid temperature difference into a heat flux. Mathematically it is defined by Newton's law of cooling, $$h = \frac{q}{A\left(T_s - T_\infty\right)},$$ where q is the convective heat transfer rate, A the surface area, T_s the surface temperature and T_∞ the bulk fluid temperature. The factors that raise or lower h are: the flow regime and velocity (turbulent, high-velocity flow gives a thinner, more disrupted boundary layer and a much higher h than still air or free convection); the fluid's own properties (thermal conductivity, viscosity, density, specific heat — captured together in the Prandtl number); the geometry and characteristic dimension of the surface (through the Reynolds/Nusselt-number correlation used); and whether the process also involves a phase change at the surface (boiling and condensing heat transfer coefficients are typically an order of magnitude higher than single-phase convection). h can be neglected — i.e. the surface can be treated as instantly at the fluid temperature — only when the internal (conductive) resistance of the food dominates the total resistance, which is exactly the high-Biot-number limit $$\mathrm{Bi} = \frac{hL_c}{k} \gg 1$$ (a vigorously agitated water or brine bath, or a jet-impingement freezer, against a food of low k and a large characteristic dimension L_c); conversely a still-air oven or freezer on a small, high-k product usually has h as the controlling resistance and it can never be dropped there.