equality equation Ex = f in situ data mass balance inequality equation literature data physiological constraints Gx h≥ linear functions numerical data 0 11 n f a b C f − ⋅ ≥ M M M food web flows the matrix equations are solved for the vector with food web flows Introduction Dynamic differential equations Steady state solutions Linear models History/Outlook
For the two spring mass example the equation of motion can be written in matrix form as For a system with two masses or more generally two degrees of freedom M and K are 2x2 matrices For a system with n degrees of freedom they are nxn matrices The spring mass system is linear A nonlinear system has more complicated equations of motion but these can always be
Now we can rewrite the principle of conservation of mass given in Equation 1 as CS dM dA dt = −•∫ρ V n 7 Rate of Rate of net inflow of mass increase of into the control volume mass M in the control volume where we have identified the physical meaning of the terms in the left and right sides of the equation In a steady state situation the time rate of change of the
Mass Balance Practice Problems F04 P1 A lake receives flow from a creek containing 10 mg/L of algae The creek flow is 100 m3/d and the lake volume is 150 m3 During the summer the algae growth constant is /d and 15% of the incoming flow evaporates and 20 m3/d infiltrates into the groundwater What is the concentration of algae in the outflow P2 A waste stream 2
This equation will help us derive balance equations for mass momentum energy Wednesday January 11 12 5 The Lagrangian Volume Problem dx 1 dt = v 1 dx 2 dt = v 2 dx dt = ua Reynolds transport theorem Relates a closed Lagrangian system moving at u b to an open Lagrangian system moving at ua Relates a closed Lagrangian system moving at u b to an
· The mass balance equation for water and sugar the mass ratio of sugar to strawberries n = 3 Plug in these numbers into the degrees of freedom equation Therefore there are no degrees of freedom and the problem is solvable Advertisement Part 3 Part 3 of 3 Finding the Solution 1 Know your goal In this case you are asked to calculate
· If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width Section 3 11 Mechanical Vibrations It s now time to take a look at an application of second order differential equations We re going to take a look at
3 D Equations of Motion These three equations express force balance in respectively the x y z directions Section Solid Mechanics Part II Kelly 7 Figure from Cauchy s Exercices de Mathematiques 1829 The Equations of Equlibrium If the material is not moving or is moving at constant velocity and is in static equilibrium then the equations of motion
differential equation for the mole fraction of species A in the fluid 1 2 A 1 AB dx C dr c D r =− Integration of this equation is straightforward and leads to the following solution 1 2 1 A AB C xr C cD r = There are two arbitrary constants that need to be evaluated Therefore we must write two boundary conditions At the surface
mass balance equation for for hydrocyclones in d Description Calculation of a steady state mass balance for 01/01/1991· This may be fitted to a function of the form S d ffi 1 FR C d FR 5 Here S d is the actual separation function for the solids size d in the feed that is recovered in the under flow product and C d is the classification function of the hydrocyclone
· An additional equation is required to solve for the three unknowns D total W final and x fin or x D avg This additional equation called the Rayleigh equation Rayleigh 1902 is derived from a differential mass balance on the light component Assume that the holdup in the column and in the accumulator is negligible Then if a differential amount of material dW of
It is required to determine density and mass content of hard phase in the suspension Solution Let us compose the equation of process material balance Q с ·ρ с = Q о ·ρ о Q ф ·ρ ф Settlement consumption Q о can be expressed through flow volumes of suspension and filtrate Q о = Q с Q ф = 10 9 5 = 0 5 m³/h Let us express and determine the density of suspension from
· The material balance equation is the simplest expression of the conservation of mass in a reservoir The equation mathematically defines the different producing mechanisms and effectively relates the reservoir fluid and rock expansion to the subsequent fluid withdrawal Contents 1 Material balance equation 2 Nomenclature 3 References 4 Noteworthy papers
∑ D = =l How determine if a differential equation is integrable and therefore holonomic • Integrable equations must be exact they must satisfy the conditions i k = 1 n jijk ki jijt i aa qq aa tq ∂∂ = ∂∂ ∂∂ = ∂∂ Key point Nonholonomic constraints do
· Mass Balance Problem The balanced equation for the synthesis of ammonia is 3 H 2 g N 2 g → 2 NH 3 g Calculate The mass in grams of NH 3 formed from the reaction of g of N 2 The mass in grams of N 2 required for form kg of
Derivation of the basic equations of fluidflows No particle in the fluid at this stage next week •Conservation of mass of the fluid •Conservation of mass of a solute applies to non sinking particles at low concentration •Conservation of momentum •Application of these basic equations to a turbulent fluid A few concepts before we get to the meat Tensor Stress
The conservation of mass equation expressed in cylindrical coordinates is given by Once again for steady flows the equation is reduced to For incompressible flows it becomes the continuity equation Stream Function For two dimensional incompressible flows the continuity equation in Cartesian coordinates is The partial differential equation still has two unknown functions u
· From the continuity equation for mass we can derive Fick s second law directly This assumes that D i is a constant which is only true for dilute solutions This is usually a good assumption for diffusion in solids diffusion of chemicals in a dilute solution water or other typical liquid solvents and diffusion of dilute trace species in the gas phase such as carbon dioxide
Analysis We take the tank as the system which is a control volume since mass crosses the boundary The mass balance for this system can be expressed as Mass balance min mout msystem mi m2 m1 2V 1V Substituting [ kg/m3] m3 kg mi 2 1 V
Begin with the species mass balance Eq 5 and the equation for the molar flux of a material of constant molar density Eq 5 Problem 9 9 Derivation of Mean Convective Concentration Derive Eqs for the constant molar flux and 9 for a representative convective concentration Cɶ i beginning with Eq 6 for the molar transport rate by parallel convection
Mass Balances Loading Rates and Fish Growth Michael B Timmons Clark Professor of Entrepreneurship Personal Enterprise Cornell University In our opinion this presentation is the most fundamental of all the short course This is where you calculate the required flows to maintain some desired level of a particular water quality parameter
Along with the Integral Mass Equation this equation can be applied to solve many problems involving finite control volumes Differential Momentum Equation The pressure surface integral in equation 3 can be converted to a volume integral using the Gradient Theorem ZZ pndAˆ = ZZZ ∇p dV The momentum flow surface integral is also similarly converted using Gauss s
Total Mass Overall Material Balance m 1 = m 2 m 3 Component A Balance m A1 = m A2 m A3 m 1 x A1 = m 2 x A2 m 3 x A3 Component B Balance m B1 = m B2 m B3 m 1 x B1 = m 2 x B2 m 3 x B3 LECTURE 9 Solving Material Balance Problems Involving Non Reactive Processes Prof Manolito E Bambase Jr Department of Chemical Engineering University of
Write the mass balance expressions for a solution that is a M in HNO2 b M in CH3NH2 c M in H3AsO4 d M in Na2HAsO4 e M in HBrO2 and M in NaBrO2 f M in NaF and saturated with CaF2 h saturated with Ag2CO3 i
This makes mass balances more complicated Need to review/introduce some ideas about reactions Stoichiometry Represents molar participation in a reaction How do you do it Like a mini mass balance Can only balance atomic species NOT total moles or moles mass of compounds Left hand side is inlet right hand side is outlet SO2 02 >SO3
Answer the following questions related to the derivation of reservoir fluid flow equations a Write the mass balance equation one dimensional one phase b List 3 commonly used expressions for relating fluid density to pressure c Write the most common relationship between velocity and pressure and write an alternative relationship used for high fluid velocities d Write the
Dividing both sides of the last equation by g0 and reversing the limits of integration yields Z2 −Z1 = Rd g0 Z p 1 p2 Tv dp p The difference Z2−Z1 is called the geopotential thickness of the layer If the virtual temperature is constant with height we get Z2 −Z1 = H Z p 1 p2 dp p = H log p1 p2 or p2 = p1exp − Z2 −Z1 H where H = RdTv
mass This law states that matter is neither created nor destroyed in the process and the total mass remains unchanged The general principle of material balance calculations is to put and solve a number of independent equations involve number of unknowns of compositions and mass flow rates of streams enter and leave the system or process The process can be
· Download Wolfram Player This Demonstration shows the solution of mass balances around a distillation column One or two feeds enter the column and a distillate stream leaves the top while a bottoms stream leaves the bottom of the column The unknown variables are in blue use sliders to select the correct values then check solution
Thus the governing equations of groundwater flow are derived using the continuity principle equation of mass balance and the Darcy law and the derivation is presented below For the derivation of differential equations for groundwater flow let us consider a small part of the aquifer called control volume having three sides of lengths ∆x ∆y and ∆z respectively Fig
Stoichiometric Equation an equation that relates the relative number of molecules or moles of reactants and products but not mass that participate in a chemical reaction To be valid the equation must be balanced For example are the following stoichiometric equations balanced C 2 H 5 OH O 2 CO 2 H 2 O NH 4 2 Cr 2 O 7 Cr 2 O 3