Bookkeeping of a Process

Abstract

Not only numerical numbers but also units are essential for quantitative expression of physical quantities. We first give a general account of the internationally standardized system of units called the International System of Units (SI). Food processing consists of processes which have been rationally designed and optimally operated based on mass and energy balances. The chapter also discusses how mass balances for steady-state and unsteady-state operations are set up. We then review some logarithmic identities as they are useful for adequately describing many phenomena occurring in food processing operations. Further, we address issues about how to draw graphs as a dominant tool for quantitatively expressing and aiding in the understanding of various phenomena.

Exercise

• 2.1 Convert the non-SI units of the following quantities to SI units.

  (a) Gas constant, R, 0.0821 L · atm/(mol · K); (b) latent heat of melting of ice, 79.7 cal/g; (c) vapor pressure of water (25 °C), 0.0323 kgf/cm²; (d) viscosity of water (25 °C), 8.90 × 10^–3 g/(cm · s); and (e) thermal conductivity of water (25 °C), 0.00145 cal/(cm · s · °C).

• 2.2 What are the wet basis and dry basis moisture contents of the fresh spaghetti made by the kneading process described in Example 2.2?

• 2.3 Dried spaghetti weighing 19.2 kg with a dry basis moisture content of 11 % was made from 20 kg of durum semolina that contains 14 % moisture on wet basis. How much out of the raw material is recovered as finished product (what is the yield)?

• 2.4 Ten kilograms of strawberries with a wet basis moisture content of 91 % is added with 5.0 kg of granulated sugar and cooked down to prepare a strawberry jam that contains 45 % moisture on wet basis. What is the weight of the finished product of strawberry jam? Granulated sugar contains negligible amount of moisture, (0.02 %) and thus assume the moisture content to be 0.

• 2.5 A drying operation is known to be able to remove 80 % of the initial moisture in a food product that contains 70 % (w/w) moisture. How much water (mass) will be evaporated if 1 kg of this food product in its initial condition is to be subjected to the same drying operation? Determine the composition (weight ratios of solid and water) of the dried food product.

• 2.6 Ten thousand kilograms (1.0 × 10⁴ kg) of sugarcane solution with a sugar concentration of 38 % (w/w) are concentrated in an evaporator to a concentration of 74 % (w/w) on a daily basis. How much sugarcane solution is produced, and how much water is evaporated in the unit of kilogram every day?

• 2.7 Tapioca starch, a raw material of noodles, confectionery, and other similar food products, is produced by drying the starch granules of cassava root from the wet basis moisture content of 66 % (w/w) to 5 % (w/w) and then pulverizing the dried granules. How much starch granules in the units of kg/h needs to be dried to produce 5000 kg/h of tapioca starch? Also, how much water is evaporated by this drying process?

• 2.8 Ten thousand kilograms (1.0 × 10⁴ kg) of saturated solution of Na₂CO₃ is prepared at 30 °C. To what temperature has one to cool down the solution from 30 °C for 3.0 × 10³ kg of Na₂CO₃ · 10H₂O crystals (the crystals do not contain water in liquid form) to crystallize out? The aqueous solubility of Na₂CO₃ is shown in Table 2.6.

  Table 2.6 Solubility of Na₂CO₃

• 2.9 A hundred tons of cellulose and 1000 kg of microorganisms are supplied to a wastewater treatment equipment on a daily basis, while 10 t of cellulose and 15,000 kg of microorganisms are discharged daily. The digestion speed of cellulose by the microorganisms is 10⁴ kg/day. The microorganisms proliferate at the speed of 2 × 10⁴ kg/day, while they perish due to lysis at the rate of 5 × 10² kg/day. Estimate the accumulation of cellulose and microorganisms within the wastewater treatment equipment.

• 2.10 Ten thousand kilograms (1.0 × 10⁴ kg) of soybeans are processed into soybean oil and soybean flakes through a three-step treatment as illustrated in Fig. 2.10. The soybeans have a composition of 34 % (w/w) protein, 27 % (w/w) carbohydrates, 10 % (w/w) fibers, 11 % (w/w) water, and 18 % (w/w) oil. In the first step of treatment, the soybeans are pulverized and pressed to separate the oil resulting in pressed oil and soybean flakes that contain 6.5 % (w/w) of oil. Assume that the pressing operation manages to separate only the oil component from the soybeans but not the other components. In the next step of treatment, the soybean flakes are subjected to solvent extraction using hexane, reducing the oil content of the soybean flakes to 0.4 % (w/w) while yielding oil-containing hexane. Assume here that there is no residue of hexane in the post-extraction soybean flakes. Finally, the flakes are dried to the moisture content of 7 % (w/w). Determine the following: (a) the weight of the soybean flakes after the first step of treatment, (b) the weight of the solvent-extracted soybean flakes after the second step of treatment, and (c) the weight of the dried soybean flakes after the third step and the protein composition.

  Fig. 2.10

  

  Three-step treatment of soybean oil processing

• 2.11 A thousand grams of water is added to 450 g of starch that initially contains 13 % water on wet basis. α-Amylase and glucoamylase (both enzymes) are added to the mixture to catalyze the hydrolysis reaction of starch, and 98 % (with reference to the glucose residues that make up the starch molecules) of the starch are broken down to glucose. What are the ratios (by weight) of the unhydrolyzed starch, water, and glucose in the sugar syrup obtained. Let the molar mass of the glucose residues that make up the starch molecules be 162 g/mol and the terminal glucose residues be also of the same molar mass.

• 2.12 Alcohol concentration is often presented in volume percent [% (v/v)]. What is the ethanol concentration in weight percent [% (w/w)] (or [wt%]) of a 12 % (v/v) ethanol solution obtained from fermentation? Let the respective densities of ethanol and water be 0.79 and 1.00 kg/L.

  The ethanol solution is separated continuously using a fractionating column (a piece equipment that leverages the differences in volatilities of the constituents of a liquid mixture for separating it into its component parts) into two ethanol solutions with respective ethanol concentrations of 20.8 and 2.5 % (w/w). The low-boiling-point fraction with the higher (ethanol) concentration is called the distillate, while the high-boiling-point fraction with the lower concentration is called the bottoms. If the 12 % (v/v) ethanol solution is supplied to the fractionating column at the rate of 100 kg/h, what are the respective output rates [kg/h] of the distillate and bottoms?

• 2.13 A mayonnaise is prepared by drizzling vegetable oil (cooking oil) into an aqueous phase consisting mainly of egg yolks and vinegar while vigorously agitating the mixture. The ratio (by weight) of vegetable oil to egg yolk to vinegar (including salt, etc., hereinafter called vinegar) is roughly 70:15:15. If we are to prepare a mayonnaise at this weight ratio with ten eggs, how much vegetable oil and vinegar do we need? Let each egg weigh 60 g and the egg yolk 20 g. At the end of the preparation process, 1250 g of mayonnaise is collected, while the rest sticks in the mixing bowl. What is the weight ratio of the unrecoverable mayonnaise?

• 2.14 Determine the pH values of hydrochloric acid solutions with concentrations of 0.01 and 0.2 mol/L. Let the degree of dissociation of hydrogen chloride be 1.

• 2.15 What is the pH of an equivolume mixture of hydrochloric acid solutions of pH 2.8 and pH 4.2? Let the degree of dissociation of hydrogen chloride be 1.

• 2.16 A sucrose solution of volume V [cm³] consists of m _W [g] of water and m _S [g] of sucrose. Assuming that the volume of the solution can be expressed as the sum of the respective volumes of sucrose and water, the relationship described by Eq. (2.42) thus holds

  $$ V=\frac{m_{\mathrm{S}}}{\rho_{\mathrm{S}}}+\frac{m_{\mathrm{W}}}{\rho_{\mathrm{W}}}\vspace*{-3pt} $$

  (2.42)

  where ρ _S [g/cm³] and ρ _W [g/cm³] are the densities of sucrose and water, respectively. Table 2.7 shows the volumes of the sucrose solutions prepared by solubilizing various quantities of sucrose in 100 g of water at a constant temperature. Find out the densities of sucrose and water.

  Table 2.7 Weight of sucrose, m _S, and the volume of its solution, V

• 2.17 A 1.0-m³ tank contains water stained with a yellow dye. Clean water is supplied into the tank at the flow rate of Q [m³/min], while the content is being discharged at the same rate. The content of the tank is measured over time for absorbance, A, at 450 nm, and the actual measured data are presented in Table 2.8. When the absorbance, A, is proportionate to the dye concentration, the relationship between time, t, and absorbance, A, can be expressed in a similar manner as Eq. (2.21) (refer to Example 2.5) by

  Table 2.8 Variation of absorbance at 450 nm of a dye solution

  $$ \frac{A}{A_0}= \exp \left(-\frac{t}{V/Q}\right) $$

  (2.43)

  where A ₀ is the initial absorbance and V is the volume of liquid in the tank. Determine the flow rate, Q, using a semilogarithmic graph paper.

• 2.18 The viscosity, μ [Pa · s], of a fluid at temperature, T [K], can be approximately expressed by

  $$ \mu =b{e}^{a/T} $$

  Table 2.9 summarizes the viscosity data of ethanol at various temperatures. Determine the parameters a and b by the use of a semilogarithmic graph paper.

  Table 2.9 Viscosity of ethanol at different temperatures

• 2.19 Activated charcoal is added to a colored sugar syrup to adsorb the color pigments. The color pigment concentration, C [g-color pigment/L-solution], and the amount of color pigment adsorbed, q [g-color pigment/g-activated charcoal], at equilibrium are shown in Table 2.10. When the relationship between C and q is described by the Freundlich equation as expressed by Eq. (2.44), determine the parameters a and b using a double logarithmic graph paper.

  Table 2.10 Adsorption of color pigments by activated charcoal

  $$ q=a{C}^b $$

  (2.44)