FUNDAMENTALS OF FOOD PROCESS ENGINEERING
CONTENTS REVIEW OF MATHEMATICAL PRINCIPLES AND APPLICATIONS IN FOOD PROCESSING 1 1 GRAPHING AND FITTING EQUATIONS TO EXPERIMENTAL DATA 1 1 1 VARIABLES AND FUNCTIONS 1 1 2 GRAPHS 1 1 3 EQUATIONS 1 1 4 LINEAR EQUATIONS 1 1 5 NONLINEAR EQUATIONS 1 2 LINEARIZATION OF NONLINEAR EQUATIONS 1 3 NONLINEAR CURVE FITTING 1 4 LOGARITHMIC AND SEMI-LOGARITHMIC GRAPHS 1 5 INTERCEPT OF LOG-LOG GRAPHS 1 6 ROOTS OF EQUATIONS 1 6 1 POLYNOMIALS 1 6 1 1 QUADRATIC EQUATION 1 6 1 2 FACTORING 1 6 1 3 ITERATION TECHNIQUE 1 7 PROGRAMMING USING VISUAL BASIC FOR APPLICATIONS IN MICROSOFT EXCEL 1 8 USE OF SPREADSHEETS TO SOLVE ENGINEERING PROBLEMS 1 9 SIMULTANEOUS EQUATIONS 1 9 1 SUBSTITUTION 1 9 2 ELIMINATION 1 9 3 DETERMINANTS 1 10 SOLUTIONS TO A SYSTEM OF LINEAR EQUATIONS USING THE “SOLVER” MACRO IN EXCEL 1 11 POWER FUNCTIONS AND EXPONENTIAL FUNCTIONS 1 12 LOGARITHMIC FUNCTIONS 1 13 DIFFERENTIAL CALCULUS 1 13 1 DE?NITION OF A DERIVATIVE 1 13 2 DIFFERENTIATION FORMULAS 1 13 3 MAXIMUM AND MINIMUM VALUES OF FUNCTIONS 1 14 INTEGRAL CALCULUS 1 14 1 INTEGRATION FORMULAS 1 14 2 INTEGRATION TECHNIQUES 1 14 2 1 CONSTANTS 1 14 2 2 INTEGRATION BY PARTS 1 14 2 3 PARTIAL FRACTIONS 1 14 2 4 SUBSTITUTION 1 15 GRAPHICAL INTEGRATION 1 15 1 RECTANGULAR RULE 1 15 2 TRAPEZOIDAL RULE 1 15 3 SIMPSON’S RULE 1 16 DIFFERENTIAL EQUATIONS 1 17 FINITE DIFFERENCE APPROXIMATION OF DIFFERENTIAL EQUATIONS PROBLEMS SUGGESTED READING UNITS AND DIMENSIONS 2 1 DE?NITION OF TERMS 2 2 SYSTEMS OF MEASUREMENT 2 3 THE SI SYSTEM 2 3 1 UNITS IN SI AND THEIR SYMBOLS 2 3 2 PRE?XES RECOMMENDED FOR USE IN SI 2 4 CONVERSION OF UNITS 2 4 1 PRECISION, ROUNDING-OFF RULE, SIGNI?CANT DIGITS 2 5 THE DIMENSIONAL EQUATION 2 6 CONVERSION OF UNITS USING THE DIMENSIONAL EQUATION 2 7 THE DIMENSIONAL CONSTANT (GC) 2 8 DETERMINATION OF APPROPRIATE SI UNITS 2 9 DIMENSIONAL CONSISTENCY OF EQUATIONS 2 10 CONVERSION OF DIMENSIONAL EQUATIONS MATERIAL BALANCES 3 1 BASIC PRINCIPLES 3 1 1 LAW OF CONSERVATION OF MASS 3 1 2 PROCESS FLOW DIAGRAMS 3 1 3 SYSTEM BOUNDARIES 3 1 4 TOTAL MASS BALANCE 3 1 5 COMPONENT MASS BALANCE 3 1 6 BASIS AND “TIE MATERIAL” 3 2 MATERIAL BALANCE PROBLEMS INVOLVED IN DILUTION, CONCENTRATION, AND DEHYDRATION 3 2 1 STEADY STATE 3 2 2 VOLUME CHANGES ON MIXING 3 2 3 CONTINUOUS VERSUS BATCH 3 2 4 RECYCLE 3 2 5 UNSTEADY STATE 3 3 BLENDING OF FOOD INGREDIENTS 3 3 1 TOTAL MASS AND COMPONENT BALANCES 3 3 2 USE OF SPECI?ED CONSTRAINTS IN EQUATIONS 3 4 MULTISTAGE PROCESSES GASES AND VAPORS 4 1 EQUATIONS OF STATE FOR IDEAL AND REAL GASES 4 1 1 THE KINETIC THEORY OF GASES 4 1 2 ABSOLUTE TEMPERATURE AND PRESSURE 4 1 3 QUANTITY OF GASES 4 1 4 THE IDEAL GAS EQUATION 4 1 4 1 P-V-T RELATIONSHIPS FOR IDEAL GASES 4 1 5 VAN DER WAAL’S EQUATION OF STATE 4 1 6 CRITICAL CONDITIONS FOR GASES 4 1 7 GAS MIXTURES 4 2 THERMODYNAMICS 4 2 1 THERMODYNAMIC VARIABLES 4 2 2 THE RELATIONSHIP BETWEEN CP AND CV FOR GASES 4 2 3 P-V-T RELATIONSHIPS FOR IDEAL GASES IN THERMODYNAMIC PROCESSES 4 2 4 CHANGES IN THERMODYNAMIC PROPERTIES, WORK, AND HEAT ASSOCIATED WITH THERMODYNAMIC PROCESSES 4 2 5 WORK AND ENTHALPY CHANGE ON ADIABATIC EXPANSION OR COMPRESSION OF AN IDEAL GAS 4 2 6 WORK AND ENTHALPY CHANGE ON ISOTHERMAL EXPANSION OR COMPRESSION OF AN IDEAL GAS 4 3 VAPOR-LIQUID EQUILIBRIUM 4 3 1 THE CLAUSIUS-CLAPEYRON EQUATION 4 3 2 LIQUID CONDENSATION FROM GAS MIXTURES ENERGY BALANCES 5 1 GENERAL PRINCIPLES 5 2 ENERGY TERMS 5 2 1 HEAT 5 2 2 HEAT CONTENT, ENTHALPY 5 2 3 SPECI?C HEAT OF SOLIDS AND LIQUIDS 5 3 ENTHALPY CHANGES IN FOODS DURING FREEZING 5 3 1 CORRELATION EQUATIONS BASED ON FREEZING POINTS OF FOOD PRODUCTS UNMODI?ED FROM THE NATURAL STATE 5 3 2 ENTHALPY CHANGES DURING THE FREEZING OF FOODS CALCULATED FROM MOLALITY OF LIQUID WATER FRACTION OF THE FOOD 5 3 3 FREEZING POINT DEPRESSION BY SOLUTES 5 3 4 AMOUNT OF LIQUID WATER AND ICE AT TEMPERATURES BELOW FREEZING 5 3 5 SENSIBLE HEAT OF WATER AND ICE AT TEMPERATURES BELOW THE FREEZING POINT 5 3 6 TOTAL ENTHALPY CHANGE 5 3 7 SPECI?C HEATS OF GASES AND VAPORS 5 4 PROPERTIES OF SATURATED AND SUPERHEATED STEAM 5 4 1 THE STEAM TABLES 5 4 1 1 THE SATURATED STEAM TABLE 5 4 1 2 THE SUPERHEATED STEAM TABLES 5 4 1 3 DOUBLE INTERPOLATION FROM SUPERHEATED STEAM TABLES 5 4 2 PROPERTIES OF STEAM HAVING LESS THAN 100% QUALITY 5 5 HEAT BALANCES FLOW OF FLUIDS 6 1 THE CONCEPT OF VISCOSITY 6 2 RHEOLOGY 6 2 1 VISCOMETRY 6 2 1 1 VISCOMETERS BASED ON FLUID FLOW THROUGH A CYLINDER 6 2 1 2 DERIVATION OF THE POISEUILLE EQUATION 6 2 1 3 VELOCITY PRO?LE AND SHEAR RATE FOR A POWER LAW FLUID 6 2 1 4 GLASS CAPILLARY VISCOMETERS 6 2 1 5 FORCED FLOW TUBE OR CAPILLARY VISCOMETRY 6 2 1 6 EVALUATION OF WALL EFFECTS IN TUBE VISCOMETRY 6 2 1 7 GLASS CAPILLARY VISCOMETER USED AS A FORCED FLOW VISCOMETER 6 2 2 EFFECT OF TEMPERATURE ON RHEOLOGICAL PROPERTIES 6 2 3 BACK EXTRUSION 6 2 4 DETERMINATION OF RHEOLOGICAL PROPERTIES OF FLUIDS USING ROTATIONAL VISCOMETERS 6 2 4 1 WIDE GAP ROTATIONAL VISCOMETER 6 2 4 2 WIDE GAP VISCOMETER WITH CYLINDRICAL SPINDLES 6 3 CONTINUOUS VISCOSITY MONITORING AND CONTROL 6 3 1 CAPILLARY VISCOMETER 6 3 2 ROTATIONAL VISCOMETER 6 3 3 VISCOSITY SENSITIVE ROTAMETER 6 4 FLOW OF FALLING FILMS 6 4 1 FILMS OF CONSTANT THICKNESS 6 4 2 TIME-DEPENDENT FILM THICKNESS 6 4 3 PROCESSES DEPENDENT ON FLUID FILM THICKNESSES 6 5 TRANSPORTATION OF FLUIDS 6 5 1 MOMENTUM BALANCE 6 5 2 THE CONTINUITY PRINCIPLE 6 6 FLUID FLOW REGIMES 6 6 1 THE REYNOLDS NUMBER 6 6 2 PIPES AND TUBES 6 6 3 FRICTIONAL RESISTANCE TO FLOW OF NEWTONIAN FLUIDS 6 6 4 FRICTIONAL RESISTANCE TO FLOW OF NON-NEWTONIAN FLUIDS 6 6 5 FRICTIONAL RESISTANCE OFFERED BY PIPE FITTINGS TO FLUID FLOW 6 7 MECHANICAL ENERGY BALANCE: THE BERNOULLI EQUATION 6 8 PUMPS 6 8 1 TYPES OF PUMPS AND THEIR CHARACTERISTICS 6 8 2 FACTORS TO BE CONSIDERED IN PUMP SELECTION 6 8 3 PERFORMANCE CURVES OF PUMPS HEAT TRANSFER 7 1 MECHANISMS OF HEAT TRANSFER 7 1 1 HEAT TRANSFER BY CONDUCTION 7 1 2 FOURIER’S FIRST LAW OF HEAT TRANSFER 7 1 3 ESTIMATION OF THERMAL CONDUCTIVITY OF FOOD PRODUCTS 7 1 4 FOURIER’S SECOND LAW OF HEAT TRANSFER 7 1 5 TEMPERATURE PRO?LE FOR UNIDIRECTIONAL HEAT TRANSFER THROUGH A SLAB 7 1 6 CONDUCTION HEAT TRANSFER THROUGH WALLS OF A CYLINDER 7 1 7 THE TEMPERATURE PRO?LE IN THE WALLS OF A CYLINDER IN STEADY-STATE HEAT TRANSFER 7 1 8 HEAT TRANSFER BY CONVECTION 7 1 8 1 NATURAL CONVECTION 7 1 8 2 FORCED CONVECTION 7 1 9 HEAT TRANSFER BY RADIATION 7 1 9 1 TYPES OF SURFACES 7 1 9 2 EFFECT OF DISTANCE BETWEEN OBJECTS ON HEAT TRANSFER 7 1 9 3 RADIANT ENERGY EXCHANGE 7 1 10 MICROWAVE AND DIELECTRIC HEATING 7 1 10 1 ENERGY ABSORPTION BY FOODS IN A MICROWAVE FIELD 7 1 10 2 RELATIVE HEATING RATES OF FOOD COMPONENTS 7 2 TEMPERATURE MEASURING DEVICES 7 2 1 LIQUID-IN-GLASS THERMOMETERS 7 2 2 FLUID-FILLED THERMOMETERS 7 2 3 BIMETALLIC STRIP THERMOMETERS 7 2 4 RESISTANCE TEMPERATURE DEVICES (RTDS) 7 2 5 THERMOCOUPLES 7 2 6 RADIATION PYROMETERS 7 2 7 ACCURATE TEMPERATURE MEASUREMENTS 7 3 STEADY-STATE HEAT TRANSFER 7 3 1 THE CONCEPT OF RESISTANCE TO HEAT TRANSFER 7 3 2 COMBINED CONVECTION AND CONDUCTION: THE OVERALL HEAT TRANSFER COEF?CIENT 7 4 HEAT EXCHANGE EQUIPMENT 7 4 1 HEAT TRANSFER IN HEAT EXCHANGERS 7 4 2 THE LOGARITHMIC MEAN TEMPERATURE DIFFERENCE 7 5 LOCAL HEAT TRANSFER COEF?CIENTS 7 5 1 DIMENSIONLESS QUANTITIES 7 5 2 EQUATIONS FOR CALCULATING HEAT TRANSFER COEF?CIENTS 7 5 2 1 SIMPLI?ED EQUATIONS FOR NATURAL CONVECTION TO AIR OR WATER 7 5 2 2 FLUIDS IN LAMINAR AND TURBULENT FLOW INSIDE TUBES 7 5 2 3 HEAT TRANSFER TO NON-NEWTONIAN FLUIDS IN LAMINAR FLOW 7 5 2 4 ADAPTING EQUATIONS FOR HEAT TRANSFER COEF?CIENTS TO NON-NEWTONIAN FLUIDS 7 6 UNSTEADY-STATE HEAT TRANSFER 7 6 1 HEATING OF SOLIDS HAVING IN?NITE THERMAL CONDUCTIVITY 7 6 2 SOLIDS WITH FINITE THERMAL CONDUCTIVITY 7 6 3 THE SEMI-IN?NITE SLAB WITH CONSTANT SURFACE TEMPERATURE 7 6 4 THE IN?NITE SLAB 7 6 5 TEMPERATURE DISTRIBUTION FOR A BRICK-SHAPED SOLID 7 6 6 USE OF HEISSLER AND GURNEY-LURIE CHARTS 7 7 CALCULATING SURFACE HEAT TRANSFER COEF?CIENTS FROM EXPERIMENTAL HEATING CURVES 7 8 FREEZING RATES SUGGESTED READINGS KINETICS OF CHEMICAL REACTIONS IN FOODS 8 1 THEORY OF REACTION RATES 8 2 TYPES OF REACTIONS 8 2 1 UNIMOLECULAR REACTIONS 8 2 2 BIMOLECULAR REACTIONS 8 2 3 REVERSIBLE REACTIONS 8 3 ENZYME REACTIONS 8 4 REACTION ORDER 8 4 1 ZERO-ORDER REACTIONS 8 4 2 FIRST-ORDER REACTIONS 8 4 3 SECOND-ORDER REACTIONS 8 4 4 NTH-ORDER REACTIONS 8 5 REACTIONS WHERE PRODUCT CONCENTRATION IS RATE LIMITING 8 6 THE REACTION RATE CONSTANT 8 7 TEMPERATURE DEPENDENCE OF REACTION RATES 8 7 1 THE ARRHENIUS EQUATION 8 7 2 THE Q10 VALUE 8 7 3 THE Z VALUE 8 8 DETERMINATION OF REACTION KINETIC PARAMETERS 8 9 USE OF CHEMICAL REACTION KINETIC DATA FOR THERMAL PROCESS OPTIMIZATION THERMAL PROCESS CALCULATIONS 9 1 PROCESSES AND SYSTEMS FOR STABILIZATION OF FOODS FOR SHELF-STABLE STORAGE: SYSTEMS REQUIREMENTS 9 1 1 IN-CAN PROCESSING 9 1 1 1 STATIONARY RETORTS 9 1 1 2 HYDROSTATIC COOKER 9 1 1 3 CONTINUOUS AGITATING RETORTS 9 1 1 4 CRATELESS RETORTS 9 1 2 PROCESSING PRODUCTS PACKAGED IN FLEXIBLE PLASTIC CONTAINERS 9 1 3 PROCESSING IN GLASS CONTAINERS 9 1 4 FLAME STERILIZATION SYSTEMS 9 1 5 CONTINUOUS FLOW STERILIZATION: ASEPTIC OR COLD FILL 9 1 6 STEAM-AIR MIXTURES FOR THERMAL PROCESSING 9 2 MICROBIOLOGICAL INACTIVATION RATES AT CONSTANT TEMPERATURE 9 2 1 RATE OF MICROBIAL INACTIVATION 9 2 2 SHAPE OF MICROBIAL INACTIVATION CURVES 9 2 3 STERILIZING VALUE OR LETHALITY OF A PROCESS 9 2 4 ACCEPTABLE STERILIZING VALUE FOR PROCESSES 9 2 5 SELECTION OF INOCULATION LEVELS IN INOCULATED PACKS 9 2 6 DETERMINATION OF D VALUES USING THE PARTIAL STERILIZATION TECHNIQUE 9 2 7 THE HEAT RESISTANCE OF SPOILAGE MICROORGANISMS 9 2 8 F0 VALUES USED IN COMMERCIAL STERILIZATION OF CANNED FOODS 9 2 9 SURFACE STERILIZATION 9 3 EFFECT OF TEMPERATURE ON THERMAL INACTIVATION OF MICROORGANISMS 9 4 INACTIVATION OF MICROORGANISMS AND ENZYMES IN CONTINUOUSLY FLOWING FLUIDS 9 4 1 TIME AND TEMPERATURE USED IN THE PASTEURIZATION OF FLUID FOODS 9 4 2 MICROBIAL INACTIVATION IN CONTINUOUSLY FLOWING FLUIDS 9 4 3 NUTRIENT DEGRADATION 9 4 4 HIGH-PRESSURE PASTEURIZATION 9 4 4 1 HIGH-PRESSURE SYSTEMS 9 4 4 2 HIGH-PRESSURE PASTEURIZATION 9 4 4 3 HIGH-PRESSURE STERILIZATION 9 4 5 STERILIZATION OF FLUIDS CONTAINING DISCREET PARTICULATES 9 5 STERILIZING VALUE OF PROCESSES EXPRESSED AS F0 9 6 THERMAL PROCESS CALCULATIONS FOR CANNED FOODS 9 6 1 THE GENERAL METHOD 9 6 2 HEAT TRANSFER EQUATIONS AND TIME-TEMPERATURE CURVES FOR CANNED FOODS 9 6 3 PLOTTING HEAT PENETRATION DATA 9 6 3 1 DETERMINATION OF FH AND J 9 6 3 2 DETERMINATION OF FC AND JC 9 6 4 FORMULA METHODS FOR THERMAL PROCESS EVALUATION 9 6 4 1 STUMBO’S PROCEDURE 9 6 4 2 HAYAKAWA’S PROCEDURE 9 6 5 EVALUATION OF PROBABILITY OF SPOILAGE FROM A GIVEN PROCESS 9 6 5 1 CONSTANT TEMPERATURE PROCESSES 9 6 5 2 PROCESS TEMPERATURE CHANGE 9 7 BROKEN HEATING CURVES 9 8 QUALITY FACTOR DEGRADATION REFRIGERATION 10 1 MECHANICAL REFRIGERATION SYSTEM 10 1 1 PRINCIPLE OF OPERATION: THE HEAT PUMP 10 1 2 REFRIGERANTS 10 1 3 THE REFRIGERATION CYCLE 10 1 4 THE REFRIGERATION CYCLE AS A SERIES OF THERMODYNAMIC PROCESSES 10 1 5 THE REFRIGERATION CYCLE ON THE PRESSURE/ENTHALPY DIAGRAM FOR A GIVEN REFRIGERANT 10 1 5 1 EXAMPLE PROBLEMS ON THE USE OF REFRIGERANT CHARTS 10 1 6 THE CONDENSER AND EVAPORATOR 10 1 6 1 PROBLEMS WITH HEAT EXCHANGE IN SYSTEMS USING ZEOTROPIC REFRIGERANTS 10 1 7 THE COMPRESSOR 10 2 REFRIGERATION LOAD 10 2 1 HEAT INCURSION THROUGH ENCLOSURES 10 2 2 HEAT INCURSION THROUGH CRACKS AND CREVICES 10 2 3 HEAT INCURSION THROUGH OPEN DOORS 10 2 4 HEAT GENERATION 10 2 5 THE UNSTEADY-STATE REFRIGERATION LOAD 10 3 COMMODITY STORAGE REQUIREMENTS 10 4 CONTROLLED ATMOSPHERE STORAGE 10 4 1 RESPIRATION 10 4 2 CA GAS COMPOSITION 10 5 MODI?ED ATMOSPHERE PACKAGING EVAPORATION 11 1 SINGLE-EFFECT EVAPORATORS 11 1 1 THE VAPOR CHAMBER 11 1 2 THE CONDENSER 11 1 3 REMOVAL OF NONCONDENSIBLE GASES 11 1 4 THE HEAT EXCHANGER 11 2 IMPROVING THE ECONOMY OF EVAPORATORS 11 2 1 VAPOR RECOMPRESSION 11 2 2 MULTIPLE-EFFECT EVAPORATORS 11 3 ENTRAINMENT 11 4 ESSENCE RECOVERY DEHYDRATION 12 1 WATER ACTIVITY 12 1 1 THERMODYNAMIC BASIS FOR WATER ACTIVITY 12 1 2 OSMOTIC PRESSURE 12 1 3 WATER ACTIVITY AT HIGH MOISTURE CONTENTS 12 1 3 1 GIBBS-DUHEM EQUATION 12 1 3 2 OTHER EQUATIONS FOR CALCULATING WATER ACTIVITY 12 1 4 WATER ACTIVITY AT LOW MOISTURE CONTENTS 12 1 4 1 THE GAB (GUGGENHEIM-ANDERSON-DE BOER) EQUATION 12 1 4 2 OTHER EQUATIONS FOR SORPTION ISOTHERMS OF FOODS 12 2 MASS TRANSFER 12 2 1 MASS DIFFUSION 12 2 2 MASS TRANSFER FROM SURFACES TO FLOWING AIR 12 3 PSYCHROMETRY 12 3 1 CARRYING CAPACITY OF GASES FOR VAPORS 12 3 2 THE PSYCHROMETRIC CHART 12 3 3 USE OF PSYCHROMETRIC CHART TO FOLLOW CHANGES IN THE PROPERTIES OF AIR-WATER MIXTURES THROUGH A PROCESS 12 4 SIMULTANEOUS HEAT AND MASS TRANSFER IN DEHYDRATION 12 5 THE STAGES OF DRYING 12 6 PREDICTION OF DRYING TIMES FROM DRYING RATE DATA 12 6 1 MATERIALS WITH ONE FALLING RATE STAGE WHERE THE RATE OF DRYING CURVE GOES THROUGH THE ORIGIN 12 6 2 MATERIALS WITH MORE THAN ONE FALLING RATE STAGE 12 6 3 THE CONSTANT DRYING RATE 12 7 SPRAY DRYING 12 7 1 DRYING TIMES IN SPRAY DRYING 12 8 FREEZE DRYING 12 8 1 DRYING TIMES FOR SYMMETRICAL DRYING 12 9 VACUUM BELT DRYER PHYSICAL SEPARATION PROCESSES 13 1 FILTRATION 13 1 1 FILTRATE FLOW THROUGH FILTER CAKE 13 1 2 CONSTANT PRESSURE FILTRATION 13 1 3 FILTRATION RATE MODEL EQUATIONS FOR PROLONGED FILTRATION WHEN FILTER CAKES EXHIBIT TIME-DEPENDENT SPECI?C RESISTANCE 13 1 4 EXPONENTIAL DEPENDENCE OF RATE ON FILTRATE VOLUME 13 1 5 MODEL EQUATION BASED ON TIME-DEPENDENT SPECI?C CAKE RESISTANCE 13 1 6 OPTIMIZATION OF FILTRATION CYCLES 13 1 7 PRESSURE-DRIVEN MEMBRANE SEPARATION PROCESSES 13 1 8 MEMBRANE SYSTEM CON?GURATIONS 13 1 9 TRANSMEMBRANE FLUX IN PRESSURE-DRIVEN MEMBRANE SEPARATION PROCESSES (POLARIZATION CONCENTRATION AND FOULING) 13 1 10 SOLUTE REJECTION 13 1 11 STERILIZING FILTRATIONS 13 1 12 ULTRA?LTRATION 13 1 13 REVERSE OSMOSIS 13 1 14 TEMPERATURE DEPENDENCE OF MEMBRANE PERMEATION RATES 13 1 15 OTHER MEMBRANE SEPARATION PROCESSES 13 2 SIEVING 13 2 1 STANDARD SIEVE SIZES 13 3 GRAVITY SEPARATIONS 13 3 1 FORCE BALANCE ON PARTICLES SUSPENDED IN A FLUID 13 3 1 1 BUOYANT FORCE 13 3 1 2 DRAG FORCE 13 3 2 TERMINAL VELOCITY 13 3 3 THE DRAG COEF?CIENT EXTRACTION 14 1 TYPES OF EXTRACTION PROCESSES SINGLE-STAGE BATCH PROCESSING MULTISTAGE CROSS-FLOW EXTRACTION MULTISTAGE COUNTERCURRENT EXTRACTION CONTINUOUS COUNTERCURRENT EXTRACTORS 14 2 GENERAL PRINCIPLES DIFFUSION SOLUBILITY EQUILIBRIUM 14 3 SOLID-LIQUID EXTRACTION: LEACHING THE EXTRACTION BATTERY: NUMBER OF EXTRACTION STAGES DETERMINATION OF THE NUMBER OF EXTRACTION STAGES USING THE PONCHON-SAVARIT DIAGRAM THE LEVER RULE IN PLOTTING POSITION OF A MIXTURE OF TWO STREAMS IN AN X-Y DIAGRAM MATHEMATICAL AND GRAPHICAL REPRESENTATION OF THE POINT J IN THE PONCHON-SAVARIT DIAGRAM MATHEMATICAL AND GRAPHICAL REPRESENTATION OF THE POINT P EQUATION OF THE OPERATING LINE AND REPRESENTATION ON THE X-Y DIAGRAM CONSTRUCTION OF THE PONCHON-SAVARIT DIAGRAM FOR THE DETERMINATION OF THE NUMBER OF IDEAL EXTRACTION STAGES 14 4 SUPERCRITICAL FLUID EXTRACTION EXTRACTION PRINCIPLES CRITICAL POINTS OF SUPERCRITICAL FLUIDS USED IN FOODS CRITICAL POINT OF MIXTURES PROPERTIES OF SUPERCRITICAL FLUIDS RELATIVE TO GASES SUPERCRITICAL FLUID EXTRACTION PARAMETERS CHAPTER VARIABLES AND FUNCTIONS A VARIABLE IS A QUANTITY THAT CAN ASSUME ANY VALUE IN ALGEBRAIC EXPRESSIONS, VARIABLES ARE REPRESENTED BY LETTERS FROM THE END OF THE ALPHABET IN PHYSICS AND ENGINEERING, ANY LETTER OF THE ALPHABET AND GREEK LETTERS ARE USED AS SYMBOLS FOR PHYSICAL QUANTITIES ANY SYMBOL MAY REPRESENT A VARIABLE IF THE VALUE OF THE PHYSICAL QUANTITY IT REPRESENTS IS NOT ?XED IN THE STATEMENT OF THE PROBLEM IN AN ALGEBRAIC EXPRESSION, THE LETTERS FROM THE BEGINNING OF THE ALPHABET OFTEN REPRESENT CONSTANTS; THAT IS, THEIR VALUES ARE ?XED THUS, IN THE EXPRESSION AX = 2BY, X AND Y REPRESENT VARIABLES AND A AND B ARE CONSTANTS A FUNCTION REPRESENTS THE MATHEMATICAL RELATIONSHIP BETWEEN VARIABLES THUS, THE TEMPERATURE IN A SOLID THAT IS BEING HEATED IN AN OVEN MAY BE EXPRESSED AS A FUNCTION OF TIME AND POSITION US- ING THE MATHEMATICAL EXPRESSION T = F(X, T) IN AN ALGEBRAIC EXPRESSION, Y = 2X + 4, Y = F(X), AND F(X) = 2X + 4 VARIABLES MAY BE DEPENDENT OR INDEPENDENT UNLESS DE?NED, THE DEPENDENT VARIABLE IN A MATHE- MATICAL EXPRESSION IS ONE THAT STANDS ALONE ON ONE SIDE OF AN EQUATION IN THE EXPRESSION Y = F(X), Y IS THE DEPENDENT AND X IS THE INDEPENDENT VARIABLE WHEN THE EXPRESSION IS REARRANGED IN THE FORM X = F(Y), X IS THE DEPENDENT AND Y IS THE INDEPENDENT VARIABLE IN PHYSICAL OR CHEMICAL SYSTEMS, THE INTERDEPENDENCE OF THE VARIABLES IS DETERMINED BY THE DESIGN OF THE EXPERIMENT THE INDEPENDENT VARIABLES ARE THOSE ?XED IN THE DESIGN OF THE EXPERIMENT, AND THE DEPENDENT VARIABLES ARE THOSE THAT ARE MEASURED FOR EXAMPLE, WHEN DETERMINING THE LOSS OF ASCORBIC ACID IN STORED CANNED FOODS, ASCORBIC ACID CONCENTRATION IS THE DEPENDENT VARIABLE AND TIME IS THE INDEPENDENT VARIABLE ON THE OTHER HAND, IF AN EXPERIMENT INVOLVES TAKING A SAMPLE OF A FOOD AND MEASURING BOTH MOISTURE CONTENT AND WATER ACTIVITY, EITHER OF THESE TWO VARIABLES MAY BE DESIGNATED AS THE DEPENDENT OR INDEPENDENT VARIABLE IN STATISTICAL DESIGN, THE TERMS “RESPONSE VARIABLE” AND “TREATMENT VARIABLE” ARE USED FOR THE DEPENDENT AND INDEPENDENT VARIABLES, RESPECTIVELY EQUATIONS AN EQUATION IS A STATEMENT OF EQUALITY EQUATIONS ARE USEFUL FOR PRESENTING EXPERIMENTAL DATA BECAUSE THEY CAN BE MATHEMATICALLY MANIPULATED FURTHERMORE, IF THE FUNCTION IS CONTINUOUS, INTERPOLATION BETWEEN EXPERIMENTALLY DERIVED VALUES FOR A VARIABLE MAY BE POSSIBLE EXPERIMENTAL DATA MAY BE ?TTED TO AN EQUATION USING ANY OF THE FOLLOWING TECHNIQUES: 1 LINEAR AND POLYNOMIAL REGRESSION: STATISTICAL METHODS ARE EMPLOYED TO DETERMINE THE COEF?- CIENTS OF A LINEAR OR POLYNOMIAL EXPRESSION INVOLVING THE INDEPENDENT AND DEPENDENT VARIABLES STATISTICAL PROCEDURES ARE BASED ON MINIMIZING THE SUM OF SQUARES FOR THE DIFFERENCE BETWEEN THE EXPERIMENTAL VALUES AND VALUES PREDICTED BY THE EQUATION 2 LINEARIZATION, DATA TRANSFORMATION, AND LINEAR REGRESSION: THE EQUATION TO WHICH THE DATA IS BEING ?TTED IS LINEARIZED THE DATA IS THEN TRANSFORMED IN ACCORDANCE WITH THE LINEARIZED EQUATION, AND A LINEAR REGRESSION WILL DETERMINE THE APPROPRIATE COEF?CIENTS FOR THE LINEARIZED EQUATION 3 GRAPHING: THE RAW OR TRANSFORMED DATA IS PLOTTED TO FORM A STRAIGHT LINE, AND FROM THE SLOPES AND INTERCEPT THE COEF?CIENTS OF THE VARIABLES IN THE EQUATION ARE DETERMINED 1 1 GRAPHING AND FITTING EQUATIONS TO EXPERIMENTAL DATA LINEAR EQUATIONS PLOTTING OF LINEAR EQUATIONS CAN BE FACILITATED BY WRITING THE EQUATION IN THE FOLLOWING FORMS: 1 THE SLOPE-INTERCEPT FORM: Y = AX + B, WHERE A = THE SLOPE, AND B = THE Y-INTERCEPT, OR THE POINT ON THE ORDINATE AT X = 0 THE SLOPE IS DETERMINED BY TAKING TWO POINTS ON THE LINE WITH COORDINATES (X1, Y1) AND (X2, Y2), AND SOLVING FOR A = (Y2 ?Y1)/(X2 ?X1) 2 THE POINT-SLOPE FORM: (Y ?B) = A (X ?C), WHERE A = SLOPE, AND B, C REPRESENT COORDINATES OF A POINT (C, B) THROUGH WHICH THE LINE MUST PASS WHEN LINEAR REGRESSION IS USED ON EXPERIMENTAL DATA, THE SLOPE AND THE INTERCEPT OF THE LINE ARE CALCULATED THE LINE MUST PASS THROUGH THE POINT THAT REPRESENTS THE MEAN OF X AND THE MEAN OF Y A LINE CAN THEN BE DRAWN EASILY USING EITHER THE POINT-SLOPE OR THE SLOPE-INTERCEPT FORMS OF THE EQUATION FOR THE LINE THE EQUATIONS FOR SLOPE AND INTERCEPT OF A LINE OBTAINED BY REGRESSION ANALYSIS OF N PAIRS OF EXPER- IMENTAL DATA ARE A = XY ? X Y /N X2 ? X 2/N ; B = Y X2 ? X XY N X2 ? X 2/N THE PROCESS OF REGRESSION INVOLVES MINIMIZING THE SQUARE OF THE DIFFERENCE BETWEEN VALUE OF Y CALCULATED BY THE REGRESSION EQUATION AND YI, THE EXPERIMENTAL VALUE OF Y IN LINEAR REGRESSION, (AX + B ?Y)2 IS CALLED THE EXPLAINED VARIATION, AND (YI ?Y)2 IS CALLED THE RANDOM ERROR OR UNEX- PLAINED VARIATION THE RATIO OF THE EXPLAINED AND UNEXPLAINED VARIATION IS CALLED THE CORRELATION COEF?CIENT IF ALL THE POINTS FALL EXACTLY ON THE REGRESSION LINE, THE VARIATION OF Y FROM THE MEAN WILL BE DUE TO THE REGRESSION EQUATION, THEREFORE EXPLAINED VARIATION EQUALS THE UNEXPLAINED VARIATION, AND THE CORRELATION COEF?CIENT IS 1 00 IF THERE IS TOO MUCH DATA SCATTER, THE RANDOM OR UNEXPLAINED VARIATION WILL BE VERY LARGE, AND THE CORRELATION COEF?CIENT WILL BE LESS THAN 1 00 THUS, REGRESSION ANALYSIS NOT ONLY DETERMINES THE EQUATION OF A LINE THAT ?TS THE DATA POINTS, BUT IT CAN ALSO BE USED TO TEST IF A PREDICTABLE RELATIONSHIP EXISTS BETWEEN THE INDEPENDENT AND DEPENDENT VARIABLES THE FORMULA FOR THE LINEAR CORRELATION COEF?CIENT IS R = N XY ? X Y N X2 ? X MUST BE TESTED FOR SIGNI?CANCE OF THE REGRESSION THE STUDENT IS REFERRED TO STATISTICS TEXTBOOKS FOR PROCEDURES TO FOLLOW IN TESTING SIGNI?CANCE OF REGRESSION FROM THE CORRELATION COEF?CIENT EXAMPLE 1 1 THE PROTEIN EF?CIENCY RATIO (PER) OF A PROTEIN IS DE?NED AS THE WEIGHT GAIN OF AN ANIMAL FED A DIET CONTAINING THE TEST PROTEIN PER UNIT WEIGHT OF PROTEIN CONSUMED DATA IS COLLECTED BY PROVIDING FEED AND WATER TO THE ANIMAL SO THE ANIMAL CAN FEED AT WILL, DETERMINING THE AMOUNT OF FEED CONSUMED, AND WEIGHING EACH ANIMAL AT DESIGNATED TIME INTERVALS THE PER MAY BE CALCULATED FROM THE SLOPE OF THE REGRESSION LINE FOR WEIGHT OF THE ANIMAL (Y) AGAINST CUMULATIVE WEIGHT OF PROTEIN CONSUMED (X) THE DATA EXPRESSED AS (X, Y) WHERE X IS THE AMOUNT OF FEED CONSUMED AND Y IS THE WEIGHT ARE AS FOLLOWS: (0, 11 5), (0, 12 2), (0, 14 0), (0, 13 3), (0, 12 5), (2 0, 16 8), (2 2, 16 7), (1 8, 15 2), (2 5, 18 4), (1 8, 16 8), (3 4, 22 8), (4 2, 22 5), (3 7, 20 7), (4 6, 25 3), (4 0, 23 5), (6 5, 28 0), (6 3, 29 5), (6 8, 31 0), (5 8, 28 5), (6 6, 29 0) PERFORM A REGRESSION ANALYSIS AND DETERMINE THE PER B = XY = ANX + BX2 + CX X2Y = ANX2 + BX3 + CX LINEARIZATION OF NONLINEAR EQUATIONS NONLINEAR EQUATIONS MAY BE LINEARIZED BY SERIES EXPANSION, BUT THE TECHNIQUE IS ONLY AN APPROXIMA- TION AND THE RESULT IS GOOD ONLY FOR A LIMITED RANGE OF VALUES FOR THE VARIABLES ANOTHER TECHNIQUE FOR LINEARIZATION INVOLVES MATHEMATICAL MANIPULATION OF THE FUNCTION AND TRANSFORMATION AND/OR GROUPING SUCH THAT THE TRANSFORMED FUNCTION ASSUMES THE FORM: F(X, Y) = AG(X, Y) + B WHERE A AND B ARE CONSTANTS WHOSE VALUES DO NOT DEPEND ON X AND Y EXAMPLE 1 2 XY = 5 LINEARIZED FORM: Y = X A PLOT OF Y AGAINST (1/X) WILL BE LINEAR EXAMPLE 1 3 Y = (Y2/X) + 4 Y2 = XY ?4X Y2 = X(Y ?4) A PLOT OF X AGAINST Y2/(Y ?4) WILL BE LINEAR EXAMPLE 1 4 THE HYPERBOLIC FUNCTION Y = 1/(B + X)