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Applied Transport Phenomena - Coggle Diagram
Applied Transport Phenomena
Introduction and course prerequisites
Problem specific information for solving
Heat/Mass transfer at interfaces
Mass
Species balance
Species equilibrium using K(partition coef)
Heat
Energy balance
Equal heat flux
Thermal equilibrium
Equal temperature
Convection BC
-kdt/dx=h(t-to)
Control volume
The volume where the physical properties are determined
Boundary conditions
Types
Dirichilet BC
BC is a fixed value
vx = 0
C=C0
T=Ts
Neumann BC
Flux or derivative is specified
Using symmetry to simplify
No flux
Please watch that the sign can change depending on the side
Use
To bridge the control volume to the external surrounding
General equations
Constitutive equations
Mathmatical relationsip that relates two or more physical properties using an modifier
Examples
Fourier/Fick
Ideal gas law
Important to decrease number of variables
Conservation equations
Energy conservation
All friction/Work is denoted as Hv
Only for isotropic materials
Mass conservation (continuity equation)
Uses density
Simplifies when density is constant
Are determined using 3D micro balances
Momemtum (navier stokes)
Needs to be conserved in all 3 directions
Chemical species conservation
Difference
Diffusion
Concentration
Reaction terms
Are all provided in the booklet
Let op: elke term heeft dezelfde eenheden in de vergelijking
Balances
Micro
Macro
Micro --> macro involves lots of math + BC
General tools
Non-dimensional numbers
Biot number
Heat resistance in object / Heat resistance surroundings
Damköhler number
Ratio between the length of the system and the distance over wich reactions occur.
Coordinate systems
Cylindrical
Use symmetry or z--> to go from cylindrical to 1d cartisian
Spherical
Cartesian
Approach solving problems
Youre job is to make a reasonable representation of the system to asnwer that specific question
Derive understanding, why/when can we ignore certain terms
We are given a physical system and asked something about it
Methods (toolbox)
Order one scaling
Scaled dimensional variable
x*=x-xr/(xc-xm)
x = physical variable
xc = maximum value in domain
x*= scaled variable
Tranformations
Stretching (xc-xm)
Translating (x-xr)
xm = minimum value in domain
Scale variable = xc-xm
Are always order(1) or approximately 0<x*<1
Must thus fit in an 1 by 1 box
if the variables in a derivative are order(1) then the derivatives are also order(1)
Only if the variables do not change rigourisly
Scaling is a non-dimensionalisation tool that brings every terms in an equation and it's bc/ic to order(1) values or less.
Application
Important notes
In proper scaling, all terms will be order(1) or less over the entire domain of interest
At least two terms must be order(1) --> dominant balance
If these conditions are not met, the system is inproperly scaled
We choose our scales and may make the wrong choice
Je wil dat variabelen en derivatives order one zijn, want kan je het probleem makkelijk oplossen met alleen de prefactors.
IC and BC must also be properly scaled (same manner)
We can choose a scale that only implies to a part of the domain
Scaling can only be done with enough context
Steps to apply scaling
3 Transform all (in)dependent variables to scaled variables
4 Determine the unknown scales by introducing the known scales
2 Write down all equations and IC/BC at play
5 Apply simplification and gain insight in the problem
Compare terms to make assumptions, term<<1 means nothing if other are also <<1
You may neglect a term that is an order of magnitude smaller than another term
1 Draw the system + make sketch to guess scales
Example
Slow reaction
Scale of x is known
Scale of c is unknown and can be derived
Fast reaction
Scale of c is known
Scale of x is unknown and can be derived
Use
Determine the minimum and maximum values of our observable quantities
Greatly simplify governing equations when in order(1) form
Determine the length and timescales governing a problem, ( first removing small terms)
Use only for processes in wich it is unknown if terms can be neglected, so not for infinitely fast or moderate processes,
Order of magnitude estimation
Sign does not matter in the order of magnitude approach
Small, x<<1 (0.1)
Order(1), 0.33<x<3
Large x>>1 (10)
Reduction in dimensionality and timescales
Timescales
Assumption types
Semi-infinite domain
Infinite over the timescale were looking for
Transient
dc/dt !=0
c = f(x,t)
When td/tp is order(1)
Pseudosteady
dc/dt = 0
c = f(x,t)
When td<<tp, so diffusion is not limiting for example
Lumped capacitance approach is then needed to solve problem
For example considering a pizza with constant termperature
No spatial variantions in conc or temp
Steady state
dc/dt = 0
c =f(x)
Characteristic types
Phenomena based, Td
Reaction timescale, td = 1/k
Diffusion timescale, td = [y]^2/D = time it takes to reach a certain distance
Dominant phenomena can change over time
Process time, Tp
Choice
One day
One year
Time we are interested in Tp=[t]
When Td/Tp is order(1), the proces time depends on this particular phenomena
Time or timescales in atp are loosely defined as timescales of diffussion, reactions, processes
Dimensionality
The number of spatial variables needed for modelling
Methods for reduction
Thinnes
When edge effects can be neglected for the general system if L>>W
Apply using scaling of conservation equation and compare sizes
Never neglect dimension with constant temperature gradient
Ignoring a part of the system
Series resistance
For processes in series at different locations, determine the rate-determining phenomena/flux
For example using the biot number
Bi <<1 = only temperature dependence on location for 1 coordinate
Derived from the boundary condition
Then describe the system using the dominant phenomena
Use mean approximation for the fin problem: mean(T(z))
Then symplify the governing equation
Fill T(z) into the governing equations
Derivative counters integral
Gradient can be determined via BC
Considerations of symmetry
Choose coordination system with smallest amount of variables
Then remove all other non-asked variables
For example the no-flux boundary condtions
Sometimes symmetry can only be made after assumptions
Gaat beiden op dezelfde manier
Methods for solving DE
Self similarity
Combined scaled variable
The scale of x is a to be determined function of time. x*= x/g(t)
g(t) can be determined using math
If the BC is not fixed in time, then the concentration must also be scaled using f(t)
Is a combination of position and time
Conditions
Applicable examples
But also convection, wave movement, processes with relation position and time
Conduction/diffussion in semi-infinite domain
Only when the process is fully discribed by rate processs and not the system dimensions. So dynamic length scales <<L
The curves must have the same shape over time
Selfsim can not be applied when the lengthscale does not go to infinity or the shape alters in time for example by reactions
Application ssim
Determine g(t)
Resultaat is telkens hetzelfde voor diffusie/conductie
Then fill the combined variable into the DE and BC using the chain rule
Determine the combined variable
First we assume ssim, by guessing the combined variables, eta = x/g(t)
Self similar solution?
Check if nothing depends on x or t
The bc/ic must collapse and not be overconstrained
Check BC/IC
In rare cases, time and space are not fully independent and can be linked mathematically. x = f(t), t = g(x). Goal is to convert a pde to a simpler ode and this method looks a bit like reduction in dimensionality
Regular pertubation theory
Use
To test the importance of a neglected term. e <<1 or e<O(1)
From going from an PDE to an solvable system of ODEs
Method
The governing equations and BC/IC are expanded to a certain order
The individual coefficients are identified(grouped in order(1), O(e), O(e2)) and solved
The taylor series of e is stated
Coefficients are substituted back into the series of e to obtain the final result.
First scale the problem and determine e
If possible, compare result with analytical solution
Definition pertubation
A deviation of a system, moving object or proces form its normal state or path, caused by an outside influence
Assumed solution
taylor series form y = y0+ey1+e^2y2+e^3y3.....
Where e is the neglected prefactor of the term
y0 = solution assuming only order one effects
y1 = solution assuming only order(e) effects, etc
Up to order(e^2) is generally sufficient for atp
Carefull: bc usually have only order(1) effects
Applications
Unidirectional flow
Unsteady state
Examples
Suddenly moving plate
kinematic viscosity nu = mu/p is the diffusion constant for momentum
Solve using scaling self-similarity
eta = y/(2vt)**0.5
BC's
vx = U at y=0
vx = 0 at y-->inf
Ld = (v
t)*
0.5 = diffusion length for momentum
Navier stokes reduces now to PDE
Changing velocity profile in time
Steady state
Examples
Flow by moving plates
Flow by a pressure gradient
Two immiscable liquids
Boundary condition l-l interface
2 shear balance, mu1dv1/dy = mu2dv2/dy
= 0 for liquid gas interface because mu(l)>>mu(g)
1 no slip condition, v1,x= v2,x
Set up NS for each phase
Time constant velocity profile
No change in velocity in the flow directions
Mean velocity U = 1/H int(vxdH)
Volumetric flow rate U *A
Entrance region
Low RE
Both viscous forces dominate
Entrance length - H
High RE
Both viscous and intertia forces(vx) are important
Use Lx >> H and substitute scaled [vy] from continuity to solve
Lx/H proportional to reynolds
Reynolds = intertia / viscous
Scale both continuity and NS
Main equations
Continuity equation
Conservation of mass
Can be used to simplify the NS equation
Can relate the velocities of different directions if scaled
Navier stokes equation
Conservation of momentum
y component
z component
x component
Derivation
dp/dt= (m
dv/dt) = m
a = F
Like to have no dependence on dimensions --> force per volume
p*dv/dt = sum(f)
Terms, x component
Left hand side
Acceleration
Temporal dvx/dt
Spatial v*nablo(v)
Partial derivatives
Right hand side
Surface forces
Pressure -dP/dx = constant, f(x)=f(t)=c
Viscous = dtyx/dy = mu*nablo^2(vx)
tyx = plane and direction force, constant for N fluid
Body forces
gravity = g
p
cos(a)
electrostatic
Has only a few exact solutions, mostly for unidirectional flow
Always start with their full form and simplify according to the problem, then scale them
Available in cartesian, spherical and cylindrical coordinates
Directionality
The number of nonzeros velocity components
Assumptions
Unidirectional
Vy = Vz = 0
Uncompressible newtonian fluid
Density is a constant
No power law in viscous forces
Infinite parralel plates
No edge effects
dvx/dz = 0
Axisymmetrical
No dependence on thetha
Nearly unidirectional flow
Lubrication approximation
Condition
Geometric
Ly/Lx = (h0-hl)/L <<1
Then one of the viscous terms can be neglected and is dp/dx>>dp/dy
So only for narrow channel
Dynamic
ReLy/Lx<<1
Then the inertia terms can be neglected
Lx and Ly are the length scale of velocity changes
Problem with a nearly unidrectional flow (so Vx>>Vy) can be solved as a unidirectional flow. But now Vx depends on x and y and dp/dx on x.
Also solve y Ns to show that pressure mainly depends on x, because [px]>>[py]
Example
Flow in a tapered channel
Volumetric flow rate is constant
Height and velocity are now functions of x
First solve for Vx, then Vy using the continuity equation and P can be determined
Forced confined heat and mass transfer in laminar flows
Forced convection
Convection driven by an external force
The convective flux is generally much larger than the diffusive flux in the same direction. So Pe>>1
Fluid velecity develops usually quicker than the concentration/temperature profile --> assume developed flow.
Fluid flow is very little altered by mass or heat transfer --> solve velocity profile first
Mass/species
Non-dimensional numbers
Schmidt
mu/pD = Pe/Re
Viscous diff rate / mass diff rate
Measure on Mass transfer entrance length / Hydrodynamic entrance length. Usually for liquids >> 1
Sherwood
Kci*L/D
Non dimensional form of the mass transfer coefficient = total mass transfer / diffussive mass transfer
Very large at z=0 and approaches a constant if z--> inf
Sh <<1 is impossible (see drawing)
Peclet
uL/D = Re*Sc
Convective flux/diffusive flux
Governing species equation
Right hand side
Generation term
Diffusive flux
Left hand side
Accumulation
Convective flux
Concentration is conserved quantity
Heat
Non dimensional numbers
Peclet
uL/a = Re*Pr
Convective flux/diffusive flux
Prandtl
mu/pa = Pe/Re
Viscous diff rate / heat diff rate
Measure on Heat transfer entrance length / Hydrodynamic entrance length.
Nusselt
Non dimensional form form of the heat transfer coefficient = total heat transfer / conductive heat transfer
h
L/k =2
h*R/k
Looks like biot number, only now is k from the fluid instead of the solid
Governing heat equation
Left hand side
Accumulation
Convective flux
Right hand side
Diffusive flux
Generation term
Temperature is conserved quantity
Application in heated tube
Entrance region
entrance length z = Pe*H
Nusselt
Nu = 2hR/k = R/d
Nu = (Pe2R/z)**1/3
h = k/d, d = boundary layer thickness
Nusselt decreases with z
Only temperature change in boundary layer
[r] = d
r* = R-r/d
Fully developed flow
Nusselt = R/d met R+-d approaches constant
Cup mixing temperature/concentration
Converts 2d PDE to 1d ODE using reduction in dimensionality
Tb = int(TVzdA)/int(VzdA) = int(TVzdA)/Q
Transfer this relation to Tb and not otherwise
dA = 2Pirdr
Can take the average without assumptions if the BC depent on the same variable
Fluid dynamics
The study of fluids and the effects of forces on fluid motion
Goals
To be able to solve problems with 20% of the work to get 80% of the answer.
To solve complicated 3D transport phenomena problems at the continuum scale approximately using simplifing assumptions and methods.
Solution strategy
Scaling
Schaal de governing equations
Neglect termen door te vergelijken
Ontdek scales door gelijk te stellen
Schaal de BC's
Vul in als nodig
Oplossen
ODE
Oplossen via bekende methode
Geeft de unscaled oplossing
PDE
Self similarity mogelijk?
Anders numeriek oplossen
Average temperature (RID)?
Voorbereiding
Wat willen ze precies weten?
Welke governing equations uit de data companion horen hierbij met welke coordinaatstelsel
Wat zijn de Boundary and intial condititions
Maak een schets van het systeem
Bedenk de scales van alle variabelen en vul zonodig in de tekening
Wat zijn de aannames en hun gevolgen
Reduction in dimensionality and timescales
Verwijder zoveel mogelijk dimensies uit de GE's
Verwijder bij steady state alle tijdsafhankelijkheid