Mass And Heat Transfer

this equation is only for one species at a time --> species i

H=h+zn^v is the volume metric fluxH is hydraulic potential or water potential over s distanceK is hydraulic conductivity-represents the ease with which fluid can be transported through a porous matrix,

φ be the volumetric porosity, (ratio of volume of void space (porevolume) to the bulk volume of a porous medium.)

analogous to diffusive heat transfer described in section 2.1D_AB diffusion coefficient --> mass diffusivity of A in B in m^2/s analgous to thermal diffusivity proportionality between diffusive mass flux and concentration gradient

where DAB is the diffusivity of A through B, T is the absolute temperature, MA and MB are molecular weights, p is absolute pressure in atm, σAB is the collision diameter in A(Omega)D,AB is a dimensionless function of the temperature and the intermolec-ular potential

Stokes Equation f=6pi(viscosity)r (mu) is medium viscosity D is macroscopic diffusion coefficient f is frictional coefficient k is boltzman constantT is absolute Temperature

cap D os ratio of transport coefficient K and differential capacity

The analogy betweenthe heat and mass transfer situations is quite straightforward. In heat transfer, energy isdiffusing as well as being lost from the surface (decaying). In mass transfer, the speciesis diffusing as well as it is decaying as a first order reaction.

m=sqrt(k"/D)

w is moisture content

makes it able to be calculated with on term aka Hieslers chart

assumptions to use hiesler chart1. Uniform initial concentration2. Constant boundary fluid concentration3. Perfect slab, cylinder or sphere4. Far from edges5. No chemical reaction (rA = 0)6. Constant diffusivity D

Dispersion depends on flow, being generally an effect of turbulent flow. Molecular diffusion, due to the thermal-kinetic energy, is always present.

studied the same way as Convective Heat TransferConvective mass transfer is the movement of mass through a medium as a result of the net motion of a material in the mediumtwo scenarios1) Convection-diffusion over a surface and 2) Convection-dispersion in a fluid or porous media.

Chapter ObjectivesUnderstand the physical processes and the rate laws describing the three modes of heat transfer, conduction, convection, and radiation.Understand the material properties that affect heat conduction in a material.Key TermDiffusion- net movement of anything down a concentration gradient(depends on random movement)Conduction-heat transfer due to concentration gradient of Temp (specific implementation of diffusion)Fourier's law- conduction equation. basically heat flux=k x Temp gradientThermal conductivity- proportionality constant between heat flow and temperature gradientThermal diffusivity- proportionality constant between Energy Flux and energy gradientHeat flow rate- amount of heat transferred per unit time( heat flux=heat flow rate/area)Advection- transport of a substance or quantity by bulk motion. Convection- heat transfer due to bulk flow of fluid over a surface(specific implementation of advection plus diffusion)Bulk flow- movement of fluid down a pressure gradient Convective heat transfer coefficient-h is a function of system geometry, fluid and flow properties, and magnitude of ΔT. Convection Equation is not law, is definition of hRadiation-from atoms emitting electromagnetic waves, doesn't require medium. Depends on surface and geometric factors

Chapter ObjectivesExplain the characteristics of electromagnetic radiation as it is incident on a surface or emitted by a surface.Calculate the magnitude and spectral quality of electromagnetic radiation emitted from a material.Explain the energy emitted by biological and environmental entities.Calculate the radiative energy exchange between two bodies.KEY TERMSelectromagnetic spectrum-reflection, absorption, and trans-mission of waves-blackbody-emission and emissivity-solar radiation-solar constant-greenhouse effect-radiative exchange-view factor-

Chapter ObjectivesIdentify the terms describing storage, convection, diffusion and generation of energy in the general governing equation for heat transfer.Specify the three common types of heat transfer boundary conditions.Describe heat transfer in a mammalian tissue with blood vessels using the bioheat transfer equation.Key Termsgoverning equation-General equation that describes heat transfer with storage, conduction, generation, convection termscontrol volume- concept of system used in governing equation, 3d system of spacestorage term- rate of change of stored energyconvection term- rate of net energy transport due to bulk flowdiffusion term- Rate of energy transport due to conductiongeneration term-Rate of generation of energyboundary temperature specified- common boundary condition needed to solve individual heat transfer problemsboundary heat flux specified-common boundary condition needed to solve individual heat transfer problemsboundary convective heat transfer-common boundary condition needed to solve individual heat transfer problemsbioheat transfer-

Chapter ObjectivesFormulate and solve for a conductive heat transfer process in a slab geometry where temperature does not change with time (steady-state).Extend the steady-state heat transfer concept for a slab to a composite slab using the overall heat transfer coefficient.Extend the steady-state heat transfer in a slab to include internal heat generation.Formulate and solve for a steady-state conductive heat transfer process in a cylindrical geometry.Key Termssteady-state-thermal resistance-R-value-thermal resistances in series-overall heat transfer coefficient-heat transfer with with energy generation-thermoregulation—behavioraland autonomic processescore body temperatureextended surfaces—fins-

CHAPTER OBJECTIVESWhere temperatures do not change with position.In a simple slab geometry where temperatures vary also with position.Near the surface of a large body (semi-infinite region).KEY TERMSinternal resistanceexternal resistanceBiot numberlumped parameter analysis1D and multi-dimensional heat conductionHeisler chartssemi-infinite region

Chapter ObjectivesExplain the process of molecular diffusion and its dependence on molecular mobility.Explain the process of capillary diffusionExplain the process of dispersion in a fluid or in a porous solid.Understand the process of convective mass transfer as due to bulk flow added to diffusion or dispersion.Explain saturated flow and unsaturated capillary flow in a porous solidHave an idea of the relative rates of the different modes of mass transfer.Explain osmotic flow.KEY TERMSdiffusion, diffusivity, and diffusion coefficientdispersion and dispersion coefficienthydraulic conductivitycapillarityosmotic flowmass and molar fluxFick’s lawDarcy’s law

Chapter ObjectivesExplain the process of convection heat transfer in a fluid over a solid surface in terms of the relatively stagnant fluid layer over the surface called the boundary layer.Calculate the convective heat transfer coefficient h knowing the flow situation.KEY TERMSconvective heat transfer coefficientvelocity boundary layerthermal boundary layerReynolds numberNusselt numberPrandtl numberGrashof numbernatural and forced convectionlaminar and turbulent flow

pa is the partial pressure of speciesxa is the concentration of the speciesH is the Henry constant ( change base on gas)

roea=ca*Maca in text can means both molar and mass concentration depending on the context unless both are used in one equation

-The common example of a solid coming in equilibrium with a gas is the potato chip typically becoming soggy when left out of the package.whereRH is the relative humidity ( in fraction)b1 and b2 are constantsand w is the equilibrium moisture content(%)

ca ,absorbed is the concentration of absorbed solute ACa is the concentration of solute A in solution K* and n are empirical constants

The F0−λT values, representing the fraction of total energy emitted by a blackbody at temperature T. Then use equation to find the fraction of energy between the two wavelengths.

Wherec = is the speed of lighth = Planck's constant (6.625E-34)κ = Boltzmann's constant (1.380E-23)T = the absolute temperature in KEb_λ,= Energy flux provided by the given wavelengthλ=wavelength

Total emissitivity ε compares the emissive power E of an actual body to the emissive power of the ideal (black) body

Monochromatic emissivity = ratio of the monochromatic emissive power of a body to the monochromatic emissive power of a blackbody *at the same wavelength and temperature

Whereq= rate of heat flowF1-2= view angle from body 1->2

Wherec=3E8 m/sv= frequency

WhereFo=the flux in W/ m^2δ =the penetration depth, distance at which the flux becomes 1/ex=distance into materialF=the flux at a location inside the material at a distance x

Functions of wavelength

Whereθ=T-Tinfinty .θb=Tb-Tinfinty .Tb=Temp at base m2=(hP)/(kAc)

BC for special case of a long fin

Following Assumptions are madeInitially all material is at freezing temperature Tm but unfrozen.All material freezes at one freezing point.Thermal conductivity of the frozen part is constant.working with a symmetric freezing of a slab

Wherex=half the thicknessTm=initial temp of materialTs=surface Temp

Whereλ=heat of fusion***********when working wih Biomaterial will give you new ΔH

WhereTs=Temp of convective fluidL=half the thicknesstf=time it takes for all the way to L to freeze

WherexA=mole fraction of A(water probably) in the solutionTA0 =freezing point of a pure liquid (K)ΔHf=the latent heat of fusionRg = gas constant ( 8.314 kJ/kmol)TA=freezing point is the freezing point of the solution in A (K)

WherexB=mole fraction of B(solute) in the solutionTA0 =freezing point of a pure liquid (K)ΔHf=the latent heat of fusionRg = gas constant ( 8.314 kJ/kmol)TA=freezing point is the freezing point of the solution in A (K)also ΔTf=TA0-TA.

WhereMA=molecular weight of waterM=molality or moles of solute/unit mass of water(or other solvent)*****************************Can use when xB<<1

mA=mass of watermB=mass of soluteMA=molecular weight of waterMB= molecular weight of solutexb=mole fraction for dilute solutions

same as in freezing but for boiling point

Wherec stands for concentration of water vaporhm=mass transfer coefficient

Whereh is the heat transfer coefficientL is the characteristic lengthk is the thermal conductivity*************For Bi < 0.1Eq h(V/A)/k < 0.1

When temperature variations are ignored. Temperature would then only vary with time.********************Suitable for large surface areas, small volumes, small convective heat transfer coefficients and large thermal conductivities

Temperature as a function of time in lumped parameter heat transfer

Where α = k/ρcp  is the thermal diffusitivity

Non-dimensional time*****************Long time is defined as Fo > 0.2. Substituten n=0

Heisler Charts has plots of relations between non-dimensional variables Wheren=x/Lm=k/hL

Ti is the constant initial temperature and Ts is the constant temperature at the two surfaces of the slab at time t > 0.******************************Temperature profile will always be symmetric

Heat Flux decreases with time

InfoGen Equation in this chapter is applied to fluids instead of solidsB/c keeping convective term, will work with equations that govern fluid flowmain goal of chapter is to find h of a fluid

Effect of the flat plate on the flow/Temp is essentially restricted to the respectively boundary layers

Whereμ=viscosityL= Characteristic Lengthuinfinty is the free stream velocity,x= distance along the flow from the leading edge *******************If Reynolds number uses x ---> becomes Rex.If Reynolds number uses L --> becomes ReL .If Reynolds number uses D---> becomes ReD.*******************Reynolds number is a flow parameter so it depends only on the properties of the fluid( ie velocity density, viscosity) also depends on the distance at which your looking atReynolds number related closely to velocity boundary layer

based on disance x along the plate

Prandtl number relates thermal boundary layer to velocity boundary layer

WhereNu= non-dimensional temperature gradient Nu compares thermal conduction in the fluid relative to the convection in the fluid.(relates entirely to the fluid and involves fluid thermal conductivity

Whereβ=Thermal expansionΔT= Temp diff( b/t surface and the bulk fluid)******************************Grashof number arises because in natural convection velocity of fluid arises from density difference************************For ideal gasesβ=(1/rho)(P/Rg*T^2) since rho=P/RgT ---> β=1/T

Characteristic Length (L) depends on the geometry of the surface over which flow takes placeWhen dimensionless number is denoted with L, it means it is the average number, whereas if it is denoted with x it is the local number

Film Temperature is the average Temp of Surface temp and temp of fluidFilm temp is important because it is the temp that you use for a given fluid property like viscosity

characteristic length L is the height of the vertical surface.

The characteristic length in the equations for horizontal plates is calculated as L =A/P, where A=the surface area P=the perimeter of the plate.

characteristic length is the diameter of the sphere( L becomes D)

characteristic length would be the height of the cylinder, L.D stands for Diameter***********************************If D is large enough compared to L can use sphere *************for a horizontal circular cylinder

WhereL=distance along the flow

characteristic length is the inner diameter of the tube

the characteristic length is the outside diameter.********************over horizontal cylinder

constants B and n are found in table to use in equation

characteristic length is the diameter of the sphere( L becomes D)

if continuity equation is applicable(du/dx=0) then u can be taken out of derivative**important to realize that the convection term is when the "CV" that our looking at has bulk flow. Different than when a surface has a convection BCUtility of the equationIt is useful for any material.It is useful for any size or shape. Similar equations can be derived for other coordinate systems.It is easier to derive the more general equation and simplifyIt is safer - as you drop terms you are aware of the reasons.Can we make it more general? (aka its limitations)To use with compressible fluids.To use when all properties vary with temperature. We need numerical solutions (explained later in the book) to solve such problems.To include mass transfer. For example, the equation cannot predict the temperature inside a steak during cooking in an oven since the equation does not include water loss from the steak.

when heat conducted out of the boundary is convected by the fluid --> the heat flux can only be conducted away as fast as it is convected away

useful in first integral to get first constant (C1)**surface heat flux is different than internal heat generation

Used in a problem where geometry and the BC are symmetric---> ex is slab of uniform thickness that is symmetrically cooledon both sidesx=0 is indicative of the line of symmetry

doesn't include bulk flow part of general equation

Whereq1-2 =the heat flow rate from 1-2 (in W or Btu/hr)A= the area normal to the direction of heat flow (m2 or ft2)T1 − T2 = the Temp diff b/t surface & fluidh=the convective heat transfer coefficient, also called the filmcoefficient. (W/m^2*C or (Btu/hr*ft^2 *F)**************Understanding h************************-This equation is not a law, but rather a defining equation for h.-h includes the conduction in the fluid in addition to bulk flow --->presence of conduction(more generally diffusion) will always exist in bulk flow-h is a function of system geometry, fluid and flow properties, and magnitude of ΔT

where k=thermal conductivity of medium (W/mC) or (W/mK)qx= rate of heat flow in the x direction (W)T= Temperature at location x (K or C)x=location in x axis qx"/A = heat flux

Whereα= thermal diffusivityρ= densitycp=specific heatk=thermal conductivityflux of energy= α x Gradient in energy (aka change in internal energy--> different from thermal conductivity because this alone does not determine temp change, also matters how much energy is need for each degree of temp change)

-ρ & cp are parts of “thermal mass” of the system.-ρ x cp=volumetric heat capacity-two type of density (important when material is porous)solid density-mass per unit volume of just the solid portion in porous mediabulk density- mass of the dried solid to its total volume(solids +pores)

Whereq=heat flow rateA=area (m^2)T=absolute Temp (K)σ=Stefan-Boltzmann Constant ( 5.676E-8 [W/m^2K^4] or .1714E-8 [Btu/hrft^2R^4 ] )