Review for Exam #1. Review of Mathematics. Weighted Mean

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1 Review for Exam #1 Review of Mathematics Weighted Mean A certain property of material 1 is P 1 and that of material is P If x 1 amount (weight or volume) of material 1 is mixed with x amount of material, what is the corresponding property of the resulting mixture? Resulting property = x1 x P1 P x1 x x1 x Note: Most (not all) properties follow the rule of weighted mean 3 1

2 Weighted Mean (contd) If the fraction of material 1,, and 3 in a product are c 1, c, and c 3 respectively and their respective properties (say density) are P 1, P, and P 3 respectively, then the resulting property of the mixtures is given by: P = c 1 (P 1 ) + c (P ) + c 3 (P 3 ) Note: c 1 + c + c 3 = 10 4 Linear Interpolation Linear interpolation involves assumption of a linear relationship between x and y If y = y 1 when x = x 1 & y = y when x = x, then for any value x 3 (that lies between x 1 & x ), the value of y is determined as follows using the concept that the slope of a straight line is the same no matter which two points are used in determining the slope: y y1 y3 y1 x x x x 1 In the above equation, all values except y 3 are known Hence, y 3 can be calculated Linear Interpolation (Graphical Visualization) y y (x, y ) y y1 y3 y1 Slope x x x x y y 1 (x 1, y 1 ) y 3 = slope (x 3 x 1 ) + y 1 x 1 x x 3 x Note: Always check to ensure that the value of y 3 always lies between the values of y 1 and y 6

3 Logarithms ln (a b) = ln (a) + ln (b) ln (a/b) = ln (a) - ln (b) Note: ln (a + b) ln (a) + ln (b) Note: ln(a) a ln ln(b) b ln (x n ) = n ln (x) ln (e x ) = x ln (e) = 1 e ln(x) = x [Log to ln: multiply by 303; ln to log: divide by 303] 7 Logarithms (Application) Consider the equation: y = x n In order to linearize the above equation and solve for n, we take the natural logarithm on both sides This results in: ln (y) = n ln (x) We can then solve for n as n = [ln(y)] / [ln(x)] 8 Units and Dimensions 9 3

4 Primary and Secondary Dimensions Primary dimensions Mass (M) Length (L) Time (T) Temperature (K) Secondary dimensions Combination of primary dimensions Example: Velocity = Distance/Time Thus, its dimensions are L/T OR LT Secondary Dimensions Force (Units: N) o Force = (mass) (acceleration) = (mass) (velocity)/(time) Dimensions: M LT -1 / T = MLT - Energy or work done (Units: Joules) o Work = (force) (distance) Dimensions: (MLT - ) (L) = ML T - Power (Units: J/s or Watts) o Power = (work)/(time) Dimensions: (ML T - )/(T) = ML T -3 Specific heat (Units: J/kg K) o Energy reqd to inc temp of 1 kg of material by 1 C Dimensions: (ML T - )/(M K) = L T - K SI Units Mass: kg (g is NOT SI unit) Length: m (cm is NOT SI unit) Time: s Temperature: K Electric current: A Amount of a substance: mol Luminous intensity: Cd 1 4

5 Conversion of Temperature 0 C = 3 F = 73 K 100 C = 1 F = 373 K -40 C = -40 F = 33 K C = ( F 3) * 5/9 F = ( C) * 9/5 + 3 K = C C change in temp = 18 F change in temp = 1 K change in temp 1 C 18 F 1 C 1 K 13 Conversion of Specific Heat 1 J/kg C = 1 J/kg K = (1/18) J/kg F 1 J/kg F = 18 J/kg C = 18 J/kg K Note 1: There is NO factor of 3 to account for as we are NOT converting temperature, but ONLY temperature change Note : Recall that specific heat was the energy required to change the temperature of unit mass of a substance by one unit 14 Mass and Energy Balances 15 5

6 Terminology (contd) Sensible heat transferred to a given mass of material (Q: J or J/s) Energy transferred from hot to cold object For a batch system: Q = m c p T Q is in J For a continuous system: Q = m c p T Q is in J/s or W Latent heat transferred to a given mass of material (Q: J or J/s) Energy transferred during phase change without any temperature change For a batch system: Q = m Q is in J For a continuous system: Q = m Q is in J/s or W Latent heat of vaporization ( vap : J/kg) Energy reqd to convert 1 kg of a liquid to vapor phase w/o temperature change It equals energy released when vapor condenses to liquid at that temperature For heating foods, the energy of condensing steam is made use of Latent heat of vaporization of water at 100 C = 5706 kj/kg Latent heat of fusion ( fus : J/kg) Energy required to convert 1 kg of a solid to liquid phase w/o temperature change Latent heat of fusion of ice at 0 C = 333 kj/kg 16 Terminology (contd) Saturation temperature and pressure Any substance can exist in more than one phase at one time For any pressure, there is a temperature at which a liquid can coexist with the vapor phase This pressure and the corresponding temperature are called the saturation pressure and saturation temperature respectively Example: Water at 1013 kpa (atmospheric pressure) has a saturation temperature of 100 C At higher pressure, saturation temperature is higher Saturated liquid and saturated vapor When only liquid phase exists at the saturation pressure and temperature, the liquid is called saturated liquid or condensate (its enthalpy is denoted by H c ) When only vapor exists at the saturation pressure and temperature, the vapor is called saturated vapor (its enthalpy is denoted by H v ) 17 Pressure Phase Diagram of Water Saturated liquid Subcooled liq Liq 1 Liq + Vap Saturated vapor Const temp 6 Vap * * * ** * 3 4 Saturated steam H c H v Enthalpy (H) 5 Superheated steam 1: Sub-cooled liquid : Saturated H v liquid 3: Sat steam (low qual) 4: Sat steam (high qual) 5: Saturated vapor 6: Superheated vapor Latent heat of vap ( vap ) at any temp or pr = H v H c Within the dome, water exists as steam, which is a mixture of liquid and vapor Here, temperature and pressure are constant and are called the saturation temperature and pressure respectively Left of dome: Subcooled liq; Within dome: Saturated steam; Right of dome: Superheated steam As we move from left to right within the dome, more and more of the water is in vapor phase The fraction that is in vapor phase is called steam quality (denoted by x ) It varies from 0 to 1 OR 0% to 100% at that temp or pr 18 6

7 Steam Quality (x) The term Steam generally refers to saturated steam and not superheated steam Steam is a mixture of liquid (condensate) and vapor The enthalpy of steam (H s ) is a weighted mean of enthalpy of condensate (H c ) and enthalpy of vapor (H v ) H s = x H v + (1 x) H c Rearranging, x = (H s H c )/(H v H c ) Note that x = 0 when H s = H c Also, x = 1 when H s = H v Higher the steam quality, higher the value of H s 0 x 1 OR 0% x 100% Note: H c & H v at saturation temperature & pressure are determined from steam tables T (Temperature in C) 0 S S to L Q versus T S: Solid (ice) L: Liquid (water) V: Vapor (superheated steam) L+V: Saturated steam Latent L L to V Latent V Q 1 = fusion(ice) = 333 kj/kg Q = c p(water) T = kj/kg Q 3 = vap(water) = 5706 kj/kg Q 1 Q Q 3 Q (Energy Supplied) 0 Energy Content Thermal energy content of a solid or liquid: m c p (T Kelvin - 73) OR m c p (T Celsius -0) This is based on the assumption that the energy content of the substance is zero at 0 C (= 73 K) Thermal energy content (or enthalpy) of steam is given by: m s (H s ) Note: H s = x (H v ) + (1 - x) H c 1 7

8 Direct Heating with Steam (Continuous System) m f, x f, c p(f), T f Cold product Direct contact HX m p, x p, c p(p), T p Warm diluted product m s, quality of x Steam Pressure, P OR Temperature, T m, x, c p, T: Mass flow rate, solids fraction, specific heat, temperature resp Subscripts: f for feed; p for product Overall mass balance: m f + m s = m p Solids balance: m f x f = m p x p Energy balance: m f {c p(f) } T f + m s (H s ) = m p {c p(p) } T p H s = (x) H v + (1 x) H c with H v and H c being determined from steam tables at the steam pressure of P or steam temperature of T Indirect Heating with Steam (Continuous System) m f, x f, c p(f), T f Cold product Indirect contact HX m p, x p, c p(p), T p Warm undiluted product m s, quality of x Steam Pressure, P OR Temperature, T 100% Condensate m, x, c p, T: Mass flow rate, solids fraction, specific heat, temperature resp Subscripts: f for feed; p for product Overall mass balance: m f = m p Solids balance: m f x f = m p x p Thus, x p = x f Energy balance: m f {c p(f) } T f + m s (H s ) = m p {c p(p) } T p + m s (H c ) H s = (x) H v + (1 x) H c with H v and H c being determined from steam tables at the absolute (not gauge) steam pressure of P or steam temperature of T 3 Indirect Cooling with Water (Continuous System) m f, x f, c p(f), T f Hot feed Indirect contact Cooler T 1 Cold water m cw m p, x p, c p(p), T p m, x, c p, T: Mass flow rate, solids fraction, specific heat, temperature resp Subscripts: f for feed; p for product; 1 for inlet water; for outlet water Overall mass balance: m f = m p Solids balance: m f x f = m p x p Thus, x p = x f Energy balance: m f {c p(f) }T f + m cw {c p(1) }T 1 = m cw {c p() }T + m p {c p(p) }T p T Cold product c p(1) and c p() are determined from tables containing the properties of water at temperatures T 1 and T respectively When approximations are used, c p(1) = c p() = 4180 J/kg K 4 8

9 Fluid Flow 5 Stress and Strain Stress: Force per unit area (Units: N/m or Pa) Strain: (Change in dimension)/(original dimension) (Units: None) Strain rate: Rate of change of strain (Units: s -1 ) Normal stress: [Normal (perpendicular) force] / [Area] Shear stress: [Shear (parallel) force] / [Area] Units: Pa Shear rate: Abbreviation for shear strain rate It is the velocity gradient (du/dx) in many cases Units: s -1 6 Flow Behavior for Time-Independent Fluids (Herschel-Bulkley Model for Shear Stress vs Shear Rate) Yield stress n 0 K( ) n = 1 n < 1 n > 1 n 0 K( ) = Shear stress (Pa) 0 = Yield stress (Pa) = Shear rate (s -1 ) K = Consistency coeff (Pa s n ) n = Flow behavior index Herschel-Bulkley Model: n 0 K( ) Newtonian 0 = 0, n = 1 Then, K = Power-law Model: n K( ) 7 9

10 Apparent Viscosity (contd) For pseudoplastic and dilatant fluids, n K For pseudoplastic fluids, app decreases with an increase in shear rate For dilatant fluids, app increases with an increase in shear rate Note: For pseudoplastic & dilatant fluids, app & do NOT change with time app / K n1 (Pseudoplastic Fluid) (Dilatant Fluid) Single point apparent viscosity: Human perception of thickness of a fluid food is correlated to app at 60 s -1 8 Rotational Viscometer (Newtonian Fluid) Principle Measure torque [a measure of shear stress (in Pa] versus rpm [a measure of shear rate (in s -1 ] T NL R i R o T: Torque (N m) N: Revolutions per second (s -1 ) L: Spindle length (m) R i, R o : Radius of spindle, cup resp (m) Plot T on y-axis versus N on x-axis The slope of this graph is 8 L/[1/R i 1/R o ] Obtain from this nistgov 9 10