1.1 Heat and Mass transfer in daily life and process/mechanical engineering Heat transfer in daily life: Heating Cooling Cooking

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1 1. Introduction 1.1 Heat and Mass transfer in daily life and process/echanical engineering Heat transfer in daily life: Heating Cooling Cooking ransfer of heat along a teperature difference fro one syste (body, ediu) to another one! Fig : Exaple cooling. Fig : Exaple heating. Fig : Exaple cooking. 1.1/

2 Heat transfer in process/echanical engineering: Heating / cooling Vaporizing Energy conversion ransfer of heat along a teperature difference fro one syste (body, ediu) to another one! Fig : Exaple heat exchanger. Fig : Exaple stea generator. Fig : Exaple vaporizer. 1.1/

3 Mass transfer in daily life: Breathing Digesting Sweating (cobined heat and ass transfer!) ransfer of ass along a potential difference fro one syste (body, ediu) to another one! Fig : Exaple breathing. Fig : Exaple digesting. Fig : Exaple sweating. 1.1/

4 Mass transfer in process / echanical engineering: reating of atter Cobining of substances Separating of substances ransfer of ass along a potential difference fro one syste (body, ediu) to another one! Fig : Exaples of processes with transfer of ass: 7: Coal filter, 8: Vacuu generator, 9: Cristallization pack, 11: Sludge filter, 12:Dialyser /

5 Requireents for heat transfer, e.g. Predefined perforance ust be achieved. Design of heating (and/or cooling) devices to avoid local overheating of heat exchanger fluids. Pressure loss ust be iniized to keep operational costs low. Efficiency of the total heat transfer process ust be optiized. Requireents for ass transfer, e.g. Predefined perforance ust be achieved. Mass flows ust be accoodated according to the properties of the substances. Efficiency of the total ass transfer process ust be optiized. Device ust be optiized with regard to costs. Suary For the design, operation and developent of heat and ass transfer in process / echanical engineering (e.g. technical stea generation, steel production and treatent, separation and cobination of atter) we have to coply with high deands with regard to efficiency, quality, perforance, costs, protection of environent. For achieving these high deands sophisticated ethods based on the basics of heat and ass transfer are applied. herefore: Soe basics of heat and ass treansfer! 1.1/

6 1.2 What is heat, ass? Heat Energy (Heat). Fro statistical therodynaics the internal energy of a syste can be expressed by the levels of energy and their occupation by the single olecules. For e.g. an ideal gas holds: E kin = 1 2 v2 = 3 2 k B ( ) ccording to equation ( ) the ean kinetic energy of the olecules is proportional to the teperature of an ideal gas (: ass of the olecules, v: velocity of the olecules, k B : Boltzann-constant, k B = 1, J K -1 ). If this is the only for of internal energy (no rotational energy, no vibrational energy, e.g. onoatoic gas), then the internal energy of the syste is proportional to teperature: U ( ) For non-ideal gases, liquids and solids the functionl dependency of internal energy on teperature is not as siple as equation ( ), but U = f() still reains. Fro equation ( ) or siilar relations for any aterial obviously teperature is necessary for quantifying energy. eperature. he teperaure is a variable of state that is unknown in echanics and has been introduced by therodynaics. Originally, teperature has been deduced fro tactile sense (soe thing feels hot or cold ). Fro therodynaics teperature is a property that is inherent for systes in theral equlibriu. Systes in theral equlibriu exhibit a coon (intensive) property which is called teperature. For easuring teperature and defining an epirical scale for teperature the dependency of appropriate properties of atter on teperature is used, e.g. teperature dependency of volue. For a linear dependency of that property on teperarature a teperature scale is given by t x = ax + b ( ) (x e.g. volue of a fluid). he constants a and b are defined with the help of easily reproducable fixpoints /

7 Celsius-Scale. If the elting point and boiling point of water are used as fixpoints (at a pressure of.1 MPa) and if these are assigned the values and 1, respectively, one obtaines fro eq. ( ): or 1 1 ; b ( ) x1 x x 1 1 x x x ( x) 1 [ C] ( ) x x a ( ) t ( ) 1 his is the teperature in the Celsius-Scale. Kelvin-Scale or therodynaic scale of teperature. If the volue V of an ideal gas is used for teperature easureent one obtaines instead of eq. ( ) (see theral equation of state of ideal gases, section ): he two fixpoints of the Celsius scale then are given by: ( ) ( V) V, p const. ( ) o V o 1 C 1 C ( ) 5) V o C o C For assigning 1 units for the range between the two fix points, equation ( ) can be rewritten as 1 o V o C 1 C ( ) 6) V o C ransforation yields:: 1 o ( ) C V o 1 C 1 V ( ) o C Fro experients one has obtained: C o C 273,16.1 [ K] ( ) o ( ) his is the teperature of the elting point of water in the Kelvin-Scale, which is shifted by 273,16 units copared to the Celsius-Scale. he elting point of water (triple point) has been fixed for the therodynaic teperature scale and has been assigned a value of 273,16 Kelvin ([K]) /

8 t present teperature is defined by international convention with the help of a nuber of fix points, which are ostly connected with phase changes of different aterials (for details see literature). n absolute teperature can be defined fro statistical therodynaics with the help of the theore of equipartition of energy, see equation ( ). Heat. If energy is transported along/as result of teperature differences (potential gradients) it is called heat. Heat, therefore, is not a state variable (sae as work) but a process variable. fter finishing the process (or change of state) heat is no longer present. With the above rigorous definition (often this definition is extended, copare section 1.3) heat as transferred energy can be quantified with the help of teperature. For that we invert the above stateent: If energy as heat Q is supplied to a syste (positive sign) or released fro a syste (negative sign), its teperature changes. For quantification of heat we consider theral equilibration of two systes of different initial teperatures ( 1 < 2 ). For the transferred heat during theral equilibration we write (heat balance, 1st law of therodynaics: conservation of energy): Q 1 c1d 2 c2d ( ) 1 2 he released and absorbed heat Q causes teperature changes in the systes (- 1 ) and ( 2 -), resp., which are inversely proportional to the ass and to the specific heat capacity c. If the specific heat capacity c is independent of teperature, we obtain a quantitative relation between heat and teperature: Q1 c1 ( 1 ) 2c2 ( 2 ) ( ) Q 1, 1, 1 before after 2, 2, 2 Fig : heral equilibration of two systes /

9 1.2.2 Mass Mass. he aount of atter of a therodynaic syste of i cheical coponents is given by its total ass [kg] and/or the ass of the single coponents i [kg]. Principle of conservation of ass. he total ass in a closed syste (closed syste: syste the boundaries of which are not pereable for ass) is constant. i const. or In conversion of coponents in a closed syste the change of ass of a coponent ust appear in an equivalent change of the ass of other coponents. his is a consequence of the principle of conservation of ass. If asses of single coponents are added to an open syste, the total ass changes accordingly. d d di ( ) d d i in diout ( ) ( ) second easure for the aount of atter of a therodynaic syste is the nuber of oles n [ol]. he aount of 1 ole containes N = 6, olecules of this atter (N : vogadro nuber, Lohschidt nuber). he unit [ol] is particularly convenient when cheical reactions occur in the syte. he ass and the nuber of oles n are siply related by M n ( ) Here M is the olar ass of the atter: M = M r.. u. N (M r : relativ olar ass, olar ass in ultiples of u, u = 1, g, atoic ass unit, 1/12 of the ass of the 12 C-isotop). Constitutive equations. In heat and ass transfer constitutive equations for the behavior of atter are needed in addition to balance equations. Soe of these will be discussed briefly in the following. heral equations of state. heral equations of state describe the behavior of atter in the for p = f(v,) oder p. V = f(p,), V: volue, p: pressure, : teperature /1 1/212

10 For gases this kind of relation can be written as polynoial in pressure. pv B p C p 2... ( ) Here V = V/n is the olar volue of the gas and the coefficients of the polynoial are called virial coefficients (therefore, eq. ( ): virial equation). For ideal gases = R., B=C=...=, so that the theral equation of state can be uniquely written as: pv R or (R: gas constant, R = 8,31441 J. ol -1. grad -1 ). he virial coefficients of non-ideal gases can be calculated fro the different fors of theral equations of state for non-ideal gases. Fro the theral equation of state according to van der Waals one obtaines: 1 pv R 1 b / V a B b R pv a RV nr ( ) ( ) Often, the copressibility factor Z is used: Z Fig : heral equation of state for ideal gases. p V R R B p C p R R... ( ) For ideal gases ( = R) the copressibility factor Z equals 1. For non-ideal gases the copes-sibility factor and theral equation of state are substance specific and can be derived fro the virial coefficients /2 4/212

11 N 2 O 2 CO 2 NH 3 CH 4 HCl a 1,39 1,36 3,592 4,17 2,253 3,667 b,3913,3183,426,377,4278,481 able : Van der Waals coefficients of soe real gases (units: a: [l 2. at. ol -2 ], b: [l. ol -1 ]). Fig : Van der Waals equation of state for NH 3. For condensed phases the relation V = f(p,) is expressed with the help of the isotheral copressibility k and the cubic theral expansion coefficient a. 1 k V V p and 1 a V V p ( ) For condensed phases (liquid and solids) generally 1-6 MPa -1 < k < 1-3 MPa -1 and a ~ 1-6 K -1. For changes of volue of condensed phases with pressure and/or teperature equation ( ) can be integrated as: V V e k V V e ( p p ) ( ) k ( p p ) V 1 a( ) ( ) a ( ) V 1 p Basicly, for condensed phases for oderate pressure and teperature changes one obtaines linear relations between volue and teperature or pressure (see easuring teperature) /3 1/212

12 Copare ideal gases: 1 V 1 1 k O(1 MPa ) V p p Vapor pressure of pure liquids. he vapor pressure of pure liquids is given by the equation of Clausius-Clapeyron: dp d Dh vap 1 V o a O( K ) at C V 273 p ( V V gas liq ) ( ) at,1 MPa he vapor pressure of a pure liquid varies exponentially with teperature. he dependency scales with the heat of vaporization. Equation ( ) often is written according to an epirical law (ntoine equation): B log p ( ) log p With V gas >> V liq and V gas = R/p it follows (Dh vap : heat of vaporization): Dh 1 dp vap dln vap, 2 d R R p p 1 d( ) Integration yields: p Dh vap 1 1 ln ( ) p R Dh ( ) ( ) 1/ Fig : Vapor pressure of a pure liquid (scheatically) /4 1/212

13 Vapor pressure of ixtures of liquids. he vapor pressure above ixtures of liquids is given by Raoult s law (ideal ixtures, exactly valid in the liit of y 1, y i : ole fraction liq iliq of coponent i in the liquid): p i y i liq p i ( ) For a binary ixture with the coponents and B one obtaines with the help of Raoult s law: p p p, p i B p B y y B liq liq p p B ( 1 y ) p liq B = const. p Fig : Raoult s law for an ideal binary ixture. ( ) 1 p y liq different representation can be given with the help of the boiling lines, = f(y ) for p = liq const, and condenion lines, = f(y ) for gas p = const. liquid or ixture of liquids is boiling, if the vapor pressure equals the (outer) total pressure. Fro equation ( ) we obtain: p pb y liq ( ) p p B Using ntoine s equation for the vapor pressure of the pure coponents yields: y liq e B B p e B / e B / B B B / ( ) Fro that we derive the requested relation = f(y ) for p = const. (boiling line). Siilarly we liq find the coposition of the gas phase as: p y gas p e B B B / y y e B / liq B / B BB / ( e e liq gain, we ay derive the requested relation = f(y ) for p = const. (condenion line). gas ) ( ) 1.2.2/5 4/211

14 B p = const. y gas y gas 1 y liq y liq Fig : Boiling and condenion lines. Fig : Equilibriu curve for an ideal binary ixture, separation factor constant. With a = p ()/p () (separation factor) we B obtain fro equation ( ): y gas a y 1 y liq liq ( a 1) ( ) Equation ( ) is called equilibriu curve y = f(y ) for p = const. gas liq he boiling behavior of liquids and liquid ixtures generally is uch ore coplex than given by the discussion of ideal binary ixtures, see literature (e.g. ulti coponent, ixtures, non-ideal ixtures, azeotropic points, ixing gaps etc.) 1.2.2/6 4/211

15 Solubility of gases in liquids. he solubility of gases in liquids is given by Henry s law (ideal ixtures, exactly valid in the liit of y, i liq y i : ole fraction of coponent i in the liquid): liq p i = const. Henry s law p p i k i y i liq ( ) If the solution exhibits ideal behavior over the entire range of y i liq then the Henry-constant k i would be the vapor pressure of the pure coponent i. his is generally not the case. Ideal behavior is found in highly diluted solutions or approxiately pure substances so that Raoult s and Henry s law constitute liiting cases for highly diluted solutions, copare figure [K] NH 3 Cl 2 H 2 S SO 2 CO 2 CH 4 O 2 H Raoult s law y liq 1 Fig : Henry s law and Raoult s as liiting cases for highly diluted solutions. able : k i [bar] for soe gases in water /7 4/211

16 1.3 ransport of energy (heat) and ass Energy (heat) and ass are properties tied to atter. herefore, transport of energy (heat) and ass are connected to the transport of atter. ransport of atter then is one way of transport of energy and ass (without transport of atter: e.g. heat transfer by radiation) ransport by convection Matter ay be thought of existing of a nuber i of ass points D which ove in transport by convection with the velocity v. Properties such as ass of cheical species ( k ), oentu (. v), and internal energy (U) are tied to atter. hese properties are denoted by F and f, respectively, where f eans the ass specific property [property per unit ass]. ransport of these properties is connected with transport of atter. he atter (ass) transported per unit tie by convection with the velocity v (convective ass flux) is given by: d ṁ dt =. D. DV V. kg ( ) s he flow rate is given by the product of velocity and area: 3 V v ( ) s where is an area perpendicular to the velocity v. For the convective transport of the property F (e.g. ass of cheical species, oentu, internal energy) in a flow then follows fro the transport of ass: J Fkon z D, F y x v Fig : Convective transport in a fluid flow. unit of = f f v s j F = f v Φ ( ) 1.3.1/1 1/212

17 For the convective flux density of a property F one obtains fro equation ( ): j Fkon unit of Φ = f v ( ) 2 s he convective flux or flux density of a property F is a vector parallel to the velocity vector which depends only on local properties (one-point transport processes) ransport by conduction (diffusion) In addition to the convective transport further transport echaniss are to be considered in fluid flow. Here, we will constrain the discussion to transport by conduction (diffusion). For diffusive transport of properties in a fluid gradients of physical quantities (potential gradients) are the driving forces. Diffusive transport of properties occurs without acroscopic convection of atter. he diffusive flux of a property F generally can be written as: unit of F JF = grad Y ( ) diff s Y J F diff grad Y Fig : Diffusive transport of a property F in a fluid (for siplicity onediensional). In equation ( ) is again an area perpendicular to grady, which is the forcing gradient (teperature gradient, ass density gradient, velocity gradient) for the diffusive transport. is a diffusive transport coefficient. Gradient grady and flux J F diff are anti-parallel vectors, inus sign in equation ( ). he flux density then is given by: unit of F j F = grad Y diff 2 s x ( ) 1.3.2/1 1/212

18 he diffusive flux or flux density (vector) of a property F depends on the gradients of physical quantities (two point processes). Detailed exaples later Superposition of convection and conduction (diffusion) he fluxes or flux densities of properties by convective and diffusive transport are vectors. In superposition of convection and diffusion these two vectors siply add to each other. Unit of Φ jf = f v kon 2 s Unit of j F = grad Y diff 2 s he size and direction of the resulting flux or flux density depends on the size of these vectors. For e.g. a positive gradient in x- direction the resulting flux can be in negative x- direction, if the convective flux in x-direction is saller than the diffusive flux. he resulting flux then is directed against direction of the convective flow. ( ) F ( ) j F d i ff j = Fges g r a d z Fig : Convective and diffusive transport of a property F. j j = Fkon y Y x j Fges Fdiff j F k o n = f v Unit of F f v grad Y ( ) 2 s he vectors of fluxes or flux densities by convective and diffusive transport add to the total flux or flux density (forulation of balances). In addition to convection and conduction (diffusion) other echaniss of transport exist (e.g. thero diffusion, Soret-effect, Dufour-effect, not to be discussed here) /

19 1.3.4 Balance equations for energy (heat) and ass With the help of the definition of convective and diffusive fluxes balance equations for the transport of energy (heat) and ass can be forulated. For this we define a balance volue in cartesian coordinates, see figure he fluxes J F,i of the propertyf (energy, ass, etc.) abut perpendicularly to the surfaces of the balance volue (J F,i coponents of the fluxes in x-, y- and z-direction). he balance of the property F follows fro the su over all fluxes of F through the surfaces of the balance volue, the accuulation of F within the balance volue and the conversion of F : accuulation of F = inflow of F outflow of F + conversion of F [Unit of F/s] ( ) he single ters in equation ( ) can be written as: z zdz J f,i (x) y x,y,z x y+dy xdx J f,x (x+dx) Fig : Balance volue for the forulation of balance equations. accuulati on : conversion : s Flux J F infow across ( ) Unit of Φ DV f ( ) t s Unit von Φ V ( ) s F D the sufaces of the balance j F D f v D gradyd (coponent J F Φ - outflow in Φ x direction), x( x) JF, x( x Dx) ( ) : volue : 1.3.4/ ( )

20 Expanding the fluxes of the property F into a aylor series (only linear ters): J F, x djf, x djf, x JF, x( x) JF, x Dx... Dx (1.3.4 dx dx For the coponents of the fluxes of the property F into the other directions of the coordinate syste we obtain analogous expressions using the sae approach. Suation over all fluxes in all three directions yields: ( f) DV t jf, x jf, y, D jf DxDyDz ydxdz x y z s DV ( ) ransforation results in: ( f) t ( x) J F, x F d iv j accuulation of F = ( x Dx) F s F ( ) inflow of F outflow of F z 6) DzDyDx + conversion of F z zdz J f,x (x) y x,y,z x ydy x+dx J f,x (xdx) Fig : Balance volue for the forulation of balance equations. pplying this derivation to the forulation of the balances for energy (heat) and ass the general property F in equation ( ) has to be replaced by energy U or ass. Furtherore, the conversion rates s F have to be specified according to the prevailing conversion echaniss, see following table /

21 Property F total ass: ass specific property f 1 flux density j f conversion Balance equation j =. v none div ( v ) t oentu: v v j v = vv (coponents: j uv, j vv, j wv ) s v = Sf i (Su of outer voluspecific forces) ( v ) t d iv j v f i Mass of cheical species k: k w k j k (Mechaniss: convection, diffusion..) s k (e.g. cheical reaction rates) ( w t k ) d iv j s k k internal energy: U c V j U (echaniss: convection, conduction (radiation..)) s U (e.g. work against volue forces) ( cv ) t d iv j U s U able : Balance equations for oentu, ass ad internal energy /

22 teperature Heat and ass transfer as boundary value probles of balance equations In the previous section the balance equations for ass and energy (heat) have been derived and discussed considering transport of energy and ass. he solutions of these equations (partial differential equations) yield the fields of e.g. teperature, ass fractions of cheical species (flow velocities, densities) in the systes under considerations. Different fro pure transport, heat or ass transfer is the transfer of ass and energy (heat) fro one syste to another due to differences of state variables, e.g. teperature, ass fractions of species etc. (section 1.1). Exaple: Equilibration of two bodies initially at different teperatures (see section and figure ). he teperature profiles within the two bodies can be calculated with the help of energy balances of the type of equation ( ). In this case non stationary conduction of heat occurs within the two bodies and the energy balance has to be shaped accordingly (see following sections). 1, 1 initial 2, 2 distance x Fig : Heat transfer as boundary value proble. Q However, the tie dependent heat Q(t) transferred fro syste 2 to syste 1 is given by heat conduction trough the boundary and, therefore, deterined by the teperature gradients at the boundary of syste 1 against syste 2 (see following sections). hese are obtained fro the solution of the respective balance equations! Generally, heat transfer (and ass transfer) are given by the boundary conditions of the balance equations for energy (and ass) and, therefore, heat and ass transfer ay be denoted as boundary value proble of balance equations /

23 Suary of section 1. Heat and ass transfer has been discussed referring to exaples fro every day life and process engineering. Heat has been identifyed as transfered energy and has been quantified with the help of teperature. Direction of heat transfer has been discussed. Matter has been quantified. Various constitutive laws for the behavior of atter have been discussed. Convective and diffusive transport of physical properties have been elaborated. he general forulation of balance equations for physical properties has been worked out. Heat and ass transfer have been identified as boundary condition probles for the balance equations of energy and ass. 1/