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# Exam 3 Cheat Sheet BE 332

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This 1 page Study Guide was uploaded by Rachel Streufert on Friday September 23, 2016. The Study Guide belongs to BE 332 at Michigan State University taught by d. reinhold in Summer 2015. Since its upload, it has received 4 views. For similar materials see Engr Prop of Bio Materials in BE at Michigan State University.

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Date Created: 09/23/16

Thermal conductivity - ability of a material to transmit heat Typical values (W/m-K): Food products (0.102-0.700), mammalian organs (0.200 -0.600), environmental products (wood and soil)(0.100 —2.600), water at 20C(0.598), water at 0C(0.560), ice at 0C(2.300) Depend on: porosity (within the food, within packed products), fiber orientation (e.g., difference between oak and pine), com position, temp Anderson approximation: k sis assumed to be 0.259 W/m -K: ???? = ???? ???? ▯ ▯ + ???? ▯ 1 − ???? ▯ Spells approximation: for moisture contents >50%: ???? = 0.056 + 0.57???? ▯ Estimation from composition (k ):iFat (0.18), protein (0.20), carbohydrate (0.245), air (0.025), water (0.6), ice (2.24), solids (0.26) ▯ Parallel model: ???? = ▯▯▯???? ▯ ▯ V i volume fraction ▯ ▯ ▯ ▯ Perpendicular model: = ▯▯▯ , criteria include: heterogeneous material, anisotrophic material, heat transfer perpendicular with fibers ▯ ▯ ▯ - Thermal diffusivity – Proportionality coefficient that relates the ability of a material to conduct and store heat Dependent on: wa ter content, temp, composition and porosity, many studies conclude that variation of fat, protein, and carb have very small e ffect on thermal diffusivity Not homogeneous with biological products Typical values ( m /h): Food products (1E -4-7E-4), environmen tal products (1E-4-10E-4), water at 20C(0.508E -4), water at 0C(0.481E -4), ice at 0C (0.436E -4) 2 2 Marten (1980): ???? = 0.146???? − 6???? ▯ + 0.100???? − 6???? + 0▯075???? − 6???? + 0.082▯ − 6???? , standard ▯rror of estimate: 0.0048E -6 m /s; ???? = 0.0057363???? ▯ + 0.00288 ???? + 273 ???? − 6, standard error of estimate: 0.14E -6 m /s Riedel (1969): ???? = 0.88???? − 7 + ???? ▯ − 0.88???? − 7 ???? ▯ ▯ Choi and Okos (1986): ???? = ▯▯▯???? ▯ ▯ - Specific heat - Amount of heat required to increase the temperature of one unit of mass by one unit , ???? = ???? ∗ ???? ???? − ???? ▯ ▯ ▯ Typical values: Water: Unfrozen: 4.18 kJ/kg -K, Frozen: 2.04 kJ/kg -K, Biological materials: Above freezing: 1 -4.18 kJ/kg-C; below freezing: 0.8-2.04 kJ/kg-C Oils/fats~ ½ of water, dry materials~ 1/3 – ¼ Siebel: includes 2 separate calculations, does not account for bound water, works well for high water content (>50%): Above freezing: ???? = 0▯837 + 3.348???? ; below ▯reezing: ???? = 0.837 ▯ 1.256???? ▯ Choi & Okos: 2 separate equations, does not account for bound water, works well for broad range of relative humidities, developed fo rm measurements of the specific heats of database of food products: ???? =▯4.189???? ▯ + 1.711???? +▯1.928???? + 1.▯47???? + 0.908▯ ▯ - Sensible heat- Heat exchange associated with change in temperature - Latent heat- Heat exchange associated with change in phase Latent heat of freezing is dominated by moisture content: ???? ▯▯▯▯▯▯▯▯ = 335???? ▯ Schwartzberg ( 1976): ???? ▯ ▯ ???? ▯ − ???? ▯ − ???? ▯ ????▯▯ − ???? (c =wpecific heat of water, c =Ipecific heat of ice, T FW =freezing point of water) Riedel (1978): ???? , = 334.1 + 2.05???? − 4.19 ∗ 10 ▯▯ ????▯ ▯ ▯ Apparent specific heat - Used to account for the effect of partial freezing on the specific heat of foods, incorporatelatent and sensible heat, determined by direct measurement - Vaporization (2257 kJ/kg), melting (335 kJ/kg), change temp ( 4.18 kJ/kg-K * change temp) - Enthalpy- Total energy in a thermodynamic system Amount of heat required to change temp is: Δ???? = ???? ∗ (ℎ − ℎ ) ▯ ▯ Depends on: temperature, water content, sugar content (solute effects on freezing) When temp of materials changes, the amount of heat removed or added is equal to the change in enthalpy: Δℎ = Δ???? = ???????? ▯ Δ???? + ???????? ????: ▯=latent heat of fusion of water, X w = only the fraction of water that undergoes a phase change ▯ ???? = ???? +▯???? − ???? ▯ 4.19 − 2.30???? −▯0.628???? ▯ : H=enthalpy of food, H fenthalpy of food at initial freezing temp, t=temp of food (°C), t f=initial freezing temp of food (°C), xs=mass fraction of food solids - Surface heat transfer coefficient - Proportionality coefficient related to the rate of heat flux between a fluid and a so(h) Depends on shape of object, material properties of object, material properties of the fluid surround the object, fluid flow ▯▯ Heat is conducted to the surface and then transferred across a boundary layer , = ℎ???? (???? − ???? ): T =flfid, T =sslid ▯▯ ▯ ▯ ▯ General values of h= Specific to foods: Moving air in hot -air drying or blast freezing (57 W/m2K), fl ow of low density fluids e.g. milk, water, and fruit juices through pipes (570-1040), flow of high viscosity starchy products, pastes, and suspensions (57 -570), still air in a fridge or freezer (5.7) More difficult to assess than other properties: must consider thickness of boundary la yer, direct measurement is best assessment because of many dependencies - Eutectic point - temp at which all of a solution turns to ice (for most food materials, < -30°) - Stress & strain: quantify how materials respond to physical forces ▯ 2 Stress: applied force divided by the area over which the force is applied, ???? = ,▯N/m , what is done to an object ▯▯ Strain: dimensionless deformation, relative change in length, ???? = , how the object responds ▯ Uniaxial loading: amount of deformation depe nds on: applied force, geometric size and shape Axial loading: force acts normal/perpendicular to the surface upon which it is applied, results in change in length of the ma terial Elastic behavior: Hooke’s Law: Based on ideal spring equation: ???? = −????????, F=force, x=distance, k= stiffness of spring Young’s modulus: how easy a material can be stretched or contracted, E=slope of linear portion of stress -strain curve, same as stiffness coeff from ideal spring, Hooke’s Law for elastic behavior: ???? = ???? ∗ ???? Shear stress and “rigidity”: results in “bending” or “twisting”, force acting parallel to the plane surface with which it is apied, Loading=shear, resulting stress=shear stress, resulting deformation= shear deformation (????) ▯ Shear modulus: how easily a materi al will bend or twist in response to applied shear stress, ???? = ▯, ???? = ???? ∗ tan (????), for small ????, -> ???? ≅ ???? ∗ ???? ▯ Bulk compression loading: pressure is applied from all directions resulting in decrease in volume, distance between molecules is decreasing, sig n convention: pos stress is considered pressure (volume decreases), neg stress is expansion (volume increases) Bulk modulus: characterizes how a material can withstand an elastic compression, expressed as K= “firmness”, ???? = −????( ▯▯), p= pressure in Pa, V=volume in m , K=bulk modulus in Pa ▯▯ Ductility & Brittleness: Ductility: characteristic of a material that undergoes considerable plastic deformation under tensile load before failure, brittleness: absence of any plastic deformation prior to failure Resilience: property of a material enabling it to endure high impact loads without inducting a stress in excess of the elastic limit, energy is absorbed during blow is stored and recovered when body is unloaded, measured by area under elastic portion of c urve Toughness: ability to absorb energy during plastic deformation, measure of the capacity of a material to sustain permanent def ormation, measured by area entire curve Other properties: Anisotropic - direction of force matters, inverse is isotropic; Viscoelast ic- time-dependent behavior (think muscle stretching); Organic - self-repair, adaptation to changes in mechanical demands ▯▯ Transverse strain: under uniaxial loading, there is a change in thickness/width, appearing as a bulge/depression along the si des of the sample, relative change in thickness, dependent on axial strain, ????▯= ▯ ▯ ▯▯▯▯▯▯▯▯▯▯ ▯▯▯▯▯▯ Poisson’s ratio: ???? = − ▯= , only applies to elastic range, between 0 and 0.5 (for biological materials), between-1 and 0.5 for others, ???? = 0.5: no volume change, incompressible solid; 0 < ???? < 0.5: “normal case for most materials, volume increase upon tens stress, volume ▯ ▯▯▯▯▯ ▯▯▯▯▯▯ decrease upon compression stress; −1 < ???? < 0: tensile stress results in increased cross sectional area ▯ ▯▯▯▯▯ ▯ ▯▯▯▯ ▯ ▯▯▯▯▯ Max/min shear: Step 1.) Max shear stress: ????▯▯▯ = ▯ + ???? ▯▯, Step 2.) Angle of max/min shear, 2 angles that are offset by 90°, ????????????2???? =▯−( ▯▯▯▯ ), Step 3.) Normal stresses at max shear, max shear angles still have normal stresses, x and y are equal, ????▯= ▯ ▯ ▯▯ ▯ ▯▯ ▯ ▯▯ Principle stresses: Step 1.) Find principle stresses???? = ▯ ▯± ▯ ▯ + ????▯ , Step 2.) Find the angle of the principle str esses, ????????????2???? = ▯▯ , Step 3.) Find which force is which using angle equations ▯,▯ ▯ ▯ ▯▯ ▯ ▯▯▯▯ ▯ Mohr’s circle steps: 1. Determine applied stresses, 2. Draw axes, 3. Locate center of the circle, 4. Plot ????▯, −???? ▯▯) and (????▯, ????▯▯), 5. Draw the diameter, 6. Find the radius, 7. Find ▯▯▯ , 8. Find principle stresses and corresponding angle

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