HEAT EXCHANGERSDouble pipe heat exchanger

In 1-1 counter flow exchanger, cold fluid flows inside all the tubes in 1 parallel in 1 pass. While in a 1-2, the tube side fluid flows in 2 pass The hot fluid enters the other end and forced to cross flow by bafflesLOG MEAN TEMP DIFFERENCE AND VARYING TEMP DROPConvection heat transfer

For counter current flow of two fluids

Where: Tm= mean temp differenceFor dA area heat balance for hot and cold fluid

Log mean temperature difference correction factors

Where: Thi= inlet temp of hot fluid (K)Tho= outlet of hot fluidTci= inlet of cold fluidTco= outlet of cold fluidCorrection factor (Ft) is determined graphically

1-2 HE

2-4 S & T HE

Cross-flow HE

Where: a= shell fluid mixed the other unmixedb= if both are unmixedUsing new nomenclature

For a counter-flow HE, CH>CC

Heat balance

Let: (mcp)H= CH(mcp)C= CCIf HT area is THI= TCO if CH>CCTHo= TCi if CC>CHFor a counter-flow HE with CC>CH

Actual HT

Finally rearranging for counter-flow

NTU= number of transfer unitsFor parallel form

Graph for given NTU

Where: a= counter-flowb= parallel flowFOULING FACTORSOverall U RADIATION HEAT TRANSFERAbsorptivity and black bodiesThermal radiaton falling upon a body may be either absorbed by the body as heat or reflected back into space,

Where: Blackbody- body that absorbs all radiant energy and reflect noneEmissivity ()- ratio of emissive power of a body to that of a black body (1.0)Kirchhoffs law- states that at the same temp T1, and

-holds for any black and nonblack surfacesStefan-Boltzmann law- states that the amount of heat transfer is proportional to the fourth power of temperature,-For black bodies: -For non black bodies, -= Stefan-Boltzmann constant, 5.676 x 10-8 W/m2-K4, or 0.1714 x 10-8 btu/hr-ft2-oR4Gray body- a body whose is independent wavelengthRADIATION TO A SMALL OBJECT FROM SURROUNDINGS,There is a net ration to a small objects Emitted by the small object: Absorbed by the small body: , =absorptivity of small body from surr. At T2 Net heat of absorption:

COMBINED RADIATION AND CONVECTIONHT by convection

HT by radiation

Total HT To solve for hr

ADVANCED RADIATION HT PRINCIPLES Exchange of radiation is affected by size, shape and relative orientation Gases in the intervening space also absorb TE Radiation spectrum and thermal radiation Energy transported by EM waves travel at c Body emitted EM, characterized by a broad spectrum thermal energy has =10-7 to 10-4 m Visible radiation (3.8x10-7 to 7.6x10-7 m) The body that absorbs and emits the maximum amount of energy is called a black body Planks Law and emissive power When a black body is heated to a temp T, photons are emitted from surface (a dist of energy)Planks equation relates the monochromatic emissive power () in W/m3:

For , (Wiens displacement law)

Stefan-Boltzmann law

Emissivity and Kirchhoffs law

A material is placed in an isothermal enclosure and allowed to reach the same temperature at thermal equilibriumIf G is the total amount of irradiation, the energy absorbed by the body and black body, resp:

Dividing:

But =1.0 and :

CONCEPT OF A GRAY BODY gray body- a surface for which the monochromatic properties are constant for all wavelengths Thus =constant=; =constant= Applying Kirchhoffs law: Thus and are equal even if body is not in thermal equilibrium with its surroundings Do not exist in practice but can be idealized with little errorView factors in radiotion: infinite black planes Two parallel and infinite black planes at T1 and T2 plane 1 emits radiation to plane 2, which is all absorbed plane 2 emits radiation to plane 1, which is all absorbed The net radiation from plane 1 to 2:

The fraction of radiation leaving 1 (or 2) that is intercepted by 2 (or 1) is F12 (F21)(which is 1.0), thus

View factpr in radiation: infinite gray planes plane is unobstructed in view so F12=F21=1.0 HGray bodies with area A1 and A2 and emissivities = and = Each eat emitted by 1: Absorbed by 2: Reflected by 2: Absorbed by 1: Reflected by 1: Absorbed by 2: Reflected by 2: : And so on Thus

View factors b/w black bodies

Consider two black planes of finite size Some of the radiation from surface 1 does not strike surface 2 (lost to surr) F12-fraction that us intercepted by 2 Not the case if there is a refrac- tory reradiatong walls present Solid angle ()- dimensionless quantity which is a measure of an angle in solid geometry

The intensity of radiation for a black body (IB)-the rate of radiation emitted per unit area projected ina a direction normal to surface and per unit solid angle:

If black body is a diffuse surface emitting equal intensity in all directions, I=k Integrating over all solid angles:

For any two black planes

Rate of radiant energy that leaves dA1:

Rate that leaves dA1 and arrives at dA2:

Plugging in def of d and

Rate that leaves dA2 and arrives at dA1:

Plugging in SB Law

Taking iterated integral to get total net heat flow:

Or For w/c

F12 geometric shape view factor (fraction of the total radiation that leaves A1 w/c strikes A2)Relation:

VIEW FACTORS FOR VARIOUS GEOMETRIESReciprocity relationship:

If surface A1 can only see surface A2, then F12=1.0If A1 sees a number of surfaces A2, A3, and all surfaces form an enclosure:

If A1 cannot see itself (flat or convex), F11=0View factor for parallel planes

View factor for adjacent perpendicular rectangles

View Factor when surfaces are connected by reradiating walls

View factor for gray bodies

MASS TRANSFERFICKS LAW OF MOLECULAR DIFFUSION

If pt 1 has greater conc of A than 2, the more A will diffuse to 2.Equation (Note that dCA=d(cXA)=cdXA: or EQUIMOLAR COUNTER DIFFUSION GASES

PA1>PA2, PB2>PB1 , P=kA is diffusing to the right, B is diffusing to the left

GAS CONVECTION WITH DIFFUSIONThe rate at which A passes a fixed point to the right is taken as a positive:

vad is diffusion velocity in m/sLet vM molar average velocity of the whole fluid relative to a stationary point and vA is the observed velocity by an observer at A.

Mult by cA:

Let N=total convective flux of the whole stream rel to the stationary point

For equimolar counter diffusion, convective term is zero:

A is diffusng through stagnant, non diffusing Bbenzene diffuses thru air in a tube

Thus (with c=P/RT and PA=xAP, cA/c=PA/P)

DIFFUSION THROUGH A VARYING-CROSS SECTIONAL AREA: SPHERELet so for a sphere

DIFFUSION THROUGH A VARYING-CROSS SECTIONAL AREA: CONDUIT OF NON-UNIFORM XL AREA

DIFFUSIVITY: GASESTwo bulb method

if rate of diff = rate of acc in V2

Material balance

PREDICTION OF DIFFUSIVTY OF GASESBased on Lennard-Jones potential

Where: P=absolute pressure in atm=average colision diameter=collision integral Fuller equation

Where: =sum of incremental volumesfound in Table 6.2-2can be used for non-polar gases/P-NP mixtureSchmidt Number- ratio of molecular momentum diff to molecular mass diff

Values for Gases:0.5 to 2, For liquids: 100-10000 for viscous liquidsDIFFUSIVITY: LIQUIDS

Slower than gases Determination: Diffusion in a long capillary tube and determined by the conc profile A dilute solution and a more conc soultion are placed in chambers on opposite side of a porous membrane Values available in Table 6.3-1

Prediction:Stokes-Einstein equation:

Where: = viscosity of solution=solute molar volume at normal TB (>0.5m3/kgmol)Wilke-Chang correlation

Where: = solute molar volume at normal TB (Table 6.3-2)=association parameter (water: 2.6, MethOH=1.9, EthOH=1.5, benzenem ether, heptane and others, 1.0)DIFFUSION: SOLIDSUsually obeys Ficks LawPermeability is often considered

If liquid is diffusing thru a porous solid:

Where: =tortuosity correction factor for diffusion path lengthFor gases in solids

CONVECTIVE MASS TRANSFER COEFFICIENTS Rate of molec diffusion is slow and rapid transfer is desired Fluid velocity is increased until turbulent Solute has concentration gradients from layer to layer but because eddies are present, it tends to equalize

3 regions cam be visualized: Thin viscous sub-layer-most by molec diffusion Transition or buffer region some eddies are present and MTC is um of tubulent and molec diff turbulent region mostly turbulent diffusion, conc decreased Analysis similar to molec diff For equimolar counter diffusion For liquids and gases

If yA1=CA1/c, yA2=CA2/c

TYPE OF MASS TRANSFER COEFFICIENTS

MASS TRANSFER COEFFICIENTS FOR A THRU STAGNANT, NON-DIFUSSING B

ANALOGIES B/W MASS HEAT AND MOMENTUM TRANSFER Newtons, Fouriers and Ficks are very similar Turbulent flow also are similar () Analogies to predict of one of given the others has been one of the great deals in chemical engineeringREYNOLDS ANALOGYMomentum and heat fluxes are Dividing the two, assuming that and are negligible Substituting turbulent q/A=h(T-Ti) and =svav2/2

Similar procedure is done for mass using

Valid when NSc and NPr are near 1.0 and only skin friction is presentChilton-Colburn J-Factor Analogy

Both for laminar and turbulent