Lessn 7 Review f fundamentals: Heat and Mass transfer
The bjective f this lessn is t review fundamentals f heat and mass transfer and discuss: 1. Cnductin heat transfer with gverning equatins fr heat cnductin, cncept f thermal cnductivity with typical values, intrduce the cncept f heat transfer resistance t cnductin. Radiatin heat transfer and present Planck s law, Stefan-Bltzmann equatin, expressin fr radiative exchange between surfaces and the cncept f radiative heat transfer resistance 3. Cnvectin heat transfer, cncept f hydrdynamic and thermal bundary layers, Newtn s law f cling, cnvective heat transfer cefficient with typical values, crrelatins fr heat transfer in frced cnvectin, free cnvectin and phase change, intrduce varius nn-dimensinal numbers 4. Basics f mass transfer Fick s law and cnvective mass transfer 5. Analgy between heat, mmentum and mass transfer 6. Multi-mde heat transfer, multi-layered walls, heat transfer netwrks, verall heat transfer cefficients 7. Fundamentals f heat exchangers At the end f the lessn the student shuld be able t: 1. Write basic equatins fr heat cnductin and derive equatins fr simpler cases. Write basic equatins fr radiatin heat transfer, estimate radiative exchange between surfaces 3. Write cnvectin heat transfer equatins, indicate typical cnvective heat transfer cefficients. Use crrelatins fr estimating heat transfer in frced cnvectin, free cnvectin and phase change 4. Express cnductive, cnvective and radiative heat transfer rates in terms f ptential and resistance. 5. Write Fick s law and cnvective mass transfer equatin 6. State analgy between heat, mmentum and mass transfer 7. Evaluate heat transfer during multi-mde heat transfer, thrugh multi-layered walls etc. using heat transfer netwrks and the cncept f verall heat transfer cefficient 8. Perfrm basic calculatin n heat exchangers 7.1. Intrductin Heat transfer is defined as energy-in-transit due t temperature difference. Heat transfer takes place whenever there is a temperature gradient within a system r whenever tw systems at different temperatures are brught int thermal cntact. Heat, which is energy-in-transit cannt be measured r bserved directly, but the effects prduced by it can be bserved and measured. Since heat transfer invlves transfer and/r cnversin f energy, all heat transfer prcesses must bey the first and secnd laws f thermdynamics. Hwever unlike thermdynamics, heat transfer
deals with systems nt in thermal equilibrium and using the heat transfer laws it is pssible t find the rate at which energy is transferred due t heat transfer. Frm the engineer s pint f view, estimating the rate f heat transfer is a key requirement. Refrigeratin and air cnditining invlves heat transfer, hence a gd understanding f the fundamentals f heat transfer is a must fr a student f refrigeratin and air cnditining. This sectin deals with a brief review f heat transfer relevant t refrigeratin and air cnditining. Generally heat transfer takes place in three different mdes: cnductin, cnvectin and radiatin. In mst f the engineering prblems heat transfer takes place by mre than ne mde simultaneusly, i.e., these heat transfer prblems are f multi-mde type. 7.. Heat transfer 7..1. Cnductin heat transfer: Cnductin heat transfer takes place whenever a temperature gradient exists in a statinary medium. Cnductin is ne f the basic mdes f heat transfer. On a micrscpic level, cnductin heat transfer is due t the elastic impact f mlecules in fluids, due t mlecular vibratin and rtatin abut their lattice psitins and due t free electrn migratin in slids. The fundamental law that gverns cnductin heat transfer is called Furier s law f heat cnductin, it is an empirical statement based n experimental bservatins and is given by: dt Q x = k.a. (7.1) dx In the abve equatin, Q x is the rate f heat transfer by cnductin in x-directin, (dt/dx) is the temperature gradient in x-directin, A is the crss-sectinal area nrmal t the x-directin and k is a prprtinality cnstant and is a prperty f the cnductin medium, called thermal cnductivity. The - sign in the abve equatin is a cnsequence f nd law f thermdynamics, which states that in spntaneus prcess heat must always flw frm a high temperature t a lw temperature (i.e., dt/dx must be negative). The thermal cnductivity is an imprtant prperty f the medium as it is equal t the cnductin heat transfer per unit crss-sectinal area per unit temperature gradient. Thermal cnductivity f materials varies significantly. Generally it is very high fr pure metals and lw fr nn-metals. Thermal cnductivity f slids is generally greater than that f fluids. Table 7.1 shws typical thermal cnductivity values at 300 K. Thermal cnductivity f slids and liquids vary mainly with temperature, while thermal cnductivity f gases depend n bth temperature and pressure. Fr istrpic materials the value f thermal cnductivity is same in all directins, while fr anistrpic materials such as wd and graphite the value f thermal cnductivity is different in different directins. In refrigeratin and air cnditining high thermal cnductivity materials are used in the cnstructin f heat exchangers, while lw
thermal cnductivity materials are required fr insulating refrigerant pipelines, refrigerated cabinets, building walls etc. Table 7.1. Thermal cnductivity values fr varius materials at 300 K Material Thermal cnductivity (W/m K) Cpper (pure) 399 Gld (pure) 317 Aluminum (pure) 37 Irn (pure) 80. Carbn steel (1 %) 43 Stainless Steel (18/8) 15.1 Glass 0.81 Plastics 0. 0.3 Wd (shredded/cemented) 0.087 Crk 0.039 Water (liquid) 0.6 Ethylene glycl (liquid) 0.6 Hydrgen (gas) 0.18 Benzene (liquid) 0.159 Air 0.06 General heat cnductin equatin: Furier s law f heat cnductin shws that t estimate the heat transfer thrugh a given medium f knwn thermal cnductivity and crss-sectinal area, ne needs the spatial variatin f temperature. In additin the temperature at any pint in the medium may vary with time als. The spatial and tempral variatins are btained by slving the heat cnductin equatin. The heat cnductin equatin is btained by applying first law f thermdynamics and Furier s law t an elemental cntrl vlume f the cnducting medium. In rectangular crdinates, the general heat cnductin equatin fr a cnducting media with cnstant therm-physical prperties is given by: T α τ T = x T + y T + z + q k 1 g (7.) k In the abve equatin, α = is a prperty f the media and is called as thermal ρc p diffusivity, q g is the rate f heat generatin per unit vlume inside the cntrl vlume and τ is the time. The general heat cnductin equatin given abve can be written in a cmpact frm using the Laplacian peratr, as:
1 T q g = T + (7.3) α τ k If there is n heat generatin inside the cntrl vlume, then the cnductin equatin becmes: 1 T = T (7.4) α τ If the heat transfer is steady and temperature des nt vary with time, then the equatin becmes: T = 0 (7.5) The abve equatin is knwn as Laplace equatin. The slutin f heat cnductin equatin alng with suitable initial and bundary cnditins gives temperature as a functin f space and time, frm which the temperature gradient and heat transfer rate can be btained. Fr example fr a simple case f ne-dimensinal, steady heat cnductin with n heat generatin (Fig. 7.1), the gverning equatin is given by: q x q x T x=0 = T 1 T x=l = T x Fig. 7.1. Steady 1-D heat cnductin d dx T = 0 (7.6) The slutin t the abve equatin with the specified bundary cnditins is given by: x T = T1 + ( T T1 ) L (7.7) and the heat transfer rate, Q x is given by: Q x dt = k A = dx T k A 1 T L ΔT = R cnd (7.8) where ΔT = T 1 -T and resistance t cnductin heat transfer, R cnd = (L/kA) Similarly fr ne-dimensinal, steady heat cnductin heat transfer thrugh a cylindrical wall the temperature prfile and heat transfer rate are given by:
1 1 ( 1) ( ) ln r/r T = T - (T-T ) (7.9) ln r /r dt (T 1 T) ΔT Q = = = r ka πkl (7.10) dr ln (r / r1) R cyl where r 1, r and L are the inner and uter radii and length f the cylinder and ln (r / r1) R cyl = is the heat transfer resistance fr the cylindrical wall. πlk Frm the abve discussin it is clear that the steady heat transfer rate by cnductin can be expressed in terms f a ptential fr heat transfer (ΔT) and a resistance fr heat transfer R, analgus t Ohm s law fr an electrical circuit. This analgy with electrical circuits is useful in dealing with heat transfer prblems invlving multiplayer heat cnductin and multimde heat transfer. Temperature distributin and heat transfer rates by cnductin fr cmplicated, multidimensinal and transient cases can be btained by slving the relevant heat cnductin equatin either by analytical methds r numerical methds. 7... Radiatin heat transfer: Radiatin is anther fundamental mde f heat transfer. Unlike cnductin and cnvectin, radiatin heat transfer des nt require a medium fr transmissin as energy transfer ccurs due t the prpagatin f electrmagnetic waves. A bdy due t its temperature emits electrmagnetic radiatin, and it is emitted at all temperatures. It is prpagated with the speed f light (3 x 10 8 m/s) in a straight line in vacuum. Its speed decreases in a medium but it travels in a straight line in hmgenus medium. The speed f light, c is equal t the prduct f wavelength λ and frequency ν, that is, 1 c = λν (7.11) The wave length is expressed in Angstrm (1 A = 10-10 m) r micrn (1 μm = 10-6 m). Thermal radiatin lies in the range f 0.1 t 100 μm, while visible light lies in the range f 0.35 t 0.75 μm. Prpagatin f thermal radiatin takes place in the frm f discrete quanta, each quantum having energy f E = hν (7.1) Where, h is Plank s cnstant, h = 6.65 x 10-34 Js. The radiatin energy is cnverted int heat when it strikes a bdy. The radiatin energy emitted by a surface is btained by integrating Planck s equatin ver all the wavelengths. Fr a real surface the radiatin energy given by Stefan- Bltzmann s law is: 4 Q=ε.σ.A.T s (7.13) where Q r = Rate f thermal energy emissin, W r
ε = Emissivity f the surface σ = Stefan-Bltzmann s cnstant, 5.669 X 10-8 W/m.K 4 A = Surface area, m T s = Surface Temperature, K The emissivity is a prperty f the radiating surface and is defined as the emissive pwer (energy radiated by the bdy per unit area per unit time ver all the wavelengths) f the surface t that f an ideal radiating surface. The ideal radiatr is called as a black bdy, whse emissivity is 1. A black bdy is a hypthetical bdy that absrbs all the incident (all wave lengths) radiatin. The term black has nthing t d with black clur. A white clured bdy can als absrb infrared radiatin as much as a black clured surface. A hllw enclsure with a small hle is an apprximatin t black bdy. Any radiatin that enters thrugh the hle is absrbed by multiple reflectins within the cavity. The hle being small very small quantity f it escapes thrugh the hle. The radiatin heat exchange between any tw surfaces 1 and at different temperatures T 1 and T is given by: Q =σ.a.f F (T -T ) (7.14) 4 4 1- ε A 1 where Q 1- = Radiatin heat transfer between 1 and, W F ε = Surface ptical prperty factr F A = Gemetric shape factr T 1,T = Surface temperatures f 1 and, K Calculatin f radiatin heat transfer with knwn surface temperatures invlves evaluatin f factrs F ε and F A. Analgus t Ohm s law fr cnductin, ne can intrduce the cncept f thermal resistance in radiatin heat transfer prblem by linearizing the abve equatin: (T1 T ) Q1 = (7.15) R rad where the radiative heat transfer resistance R rad is given by: T-T 1 R rad = 4 4 σafε F A(T1 -T ) (7.16) 7..3. Cnvectin Heat Transfer: Cnvectin heat transfer takes place between a surface and a mving fluid, when they are at different temperatures. In a strict sense, cnvectin is nt a basic mde f heat transfer as the heat transfer frm the surface t the fluid cnsists f tw mechanisms perating simultaneusly. The first ne is energy transfer due t mlecular mtin (cnductin) thrugh a fluid layer adjacent t the surface, which remains statinary with respect t the slid surface due t n-slip cnditin. Superimpsed upn this cnductive mde is energy transfer by the macrscpic mtin f fluid particles by virtue f an external frce, which culd be generated by a pump r fan (frced cnvectin) r generated due t buyancy, caused by density gradients.
When fluid flws ver a surface, its velcity and temperature adjacent t the surface are same as that f the surface due t the n-slip cnditin. The velcity and temperature far away frm the surface may remain unaffected. The regin in which the velcity and temperature vary frm that f the surface t that f the free stream are called as hydrdynamic and thermal bundary layers, respectively. Figure 7. shw that fluid with free stream velcity U flws ver a flat plate. In the vicinity f the surface as shwn in Figure 7., the velcity tends t vary frm zer (when the surface is statinary) t its free stream value U. This happens in a narrw regin whse thickness is f the rder f Re -0.5 L (Re L = U L/ν) where there is a sharp velcity gradient. This narrw regin is called hydrdynamic bundary layer. In the hydrdynamic bundary layer regin the inertial terms are f same rder magnitude as the viscus terms. Similarly t the velcity gradient, there is a sharp temperature gradient in this vicinity f the surface if the temperature f the surface f the plate is different frm that f the flw stream. This regin is called thermal bundary layer, δ t whse thickness is f the rder f (Re L Pr) -0.5, where Pr is the Prandtl number, given by: c p,f μ f ν f Pr = = (7.17) k α In the expressin fr Prandtl number, all the prperties refer t the flwing fluid. f f Fig. 7.. Velcity distributin f flw ver a flat plate In the thermal bundary layer regin, the cnductin terms are f same rder f magnitude as the cnvectin terms. The mmentum transfer is related t kinematic viscsity ν while the diffusin f heat is related t thermal diffusivity α hence the rati f thermal bundary layer t viscus bundary layer is related t the rati ν/α, Prandtl number. Frm the expressins fr bundary layer thickness it can be seen that the rati f thermal bundary layer thickness t the viscus bundary layer thickness depends upn Prandtl number. Fr large Prandtl numbers δ t < δ and fr small Prandtl numbers, δ t > δ. It can als be seen that as the Reynlds number increases, the bundary layers becme narrw, the temperature gradient becmes large and the heat transfer rate increases.
Since the heat transfer frm the surface is by mlecular cnductin, it depends upn the temperature gradient in the fluid in the immediate vicinity f the surface, i.e. dt Q = ka (7.18) dy y = 0 Since temperature difference has been recgnized as the ptential fr heat transfer it is cnvenient t express cnvective heat transfer rate as prprtinal t it, i.e. dt Q = k f A = h ca(tw T ) (7.19) dy y = 0 The abve equatin defines the cnvective heat transfer cefficient h c. This equatin Q = h ca(tw T ) is als referred t as Newtn s law f cling. Frm the abve equatin it can be seen that the cnvective heat transfer cefficient hc is given by: h c = k f (T dt dy w T y = 0 ) (7.0) The abve equatin suggests that the cnvective heat transfer cefficient (hence heat dt transfer by cnvectin) depends n the temperature gradient near the dy y = 0 surface in additin t the thermal cnductivity f the fluid and the temperature difference. The temperature gradient near the wall depends n the rate at which the fluid near the wall can transprt energy int the mainstream. Thus the temperature gradient depends n the flw field, with higher velcities able t pressure sharper temperature gradients and hence higher heat transfer rates. Thus determinatin f cnvectin heat transfer requires the applicatin f laws f fluid mechanics in additin t the laws f heat transfer. Table 7. Typical rder-f magnitude values f cnvective heat transfer cefficients Type f fluid and flw Cnvective heat transfer cefficient h c, (W/m K) Air, free cnvectin 6 30 Water, free cnvectin 0 100 Air r superheated steam, frced cnvectin 30 300 Oil, frced cnvectin 60 1800 Water, frced cnvectin 300 18000 Synthetic refrigerants, biling 500-3000 Water, biling 3000 60000 Synthetic refrigerants, cndensing 1500-5000 Steam, cndensing 6000 10000 Traditinally, frm the manner in which the cnvectin heat transfer rate is defined, evaluating the cnvective heat transfer cefficient has becme the main bjective f
the prblem. The cnvective heat transfer cefficient can vary widely depending upn the type f fluid and flw field and temperature difference. Table 7. shws typical rder-f-magnitude values f cnvective heat transfer cefficients fr different cnditins. Cnvective heat transfer resistance: Similar t cnductin and radiatin, cnvective heat transfer rate can be written in terms f a ptential and resistance, i.e., (Tw T ) Q = h ca(tw T ) = (7.1) R cnv where the cnvective heat transfer resistance, R cnv = 1/(h c A) Determinatin f cnvective heat transfer cefficient: Evaluatin f cnvective heat transfer cefficient is difficult as the physical phenmenn is quite cmplex. Analytically, it can be determined by slving the mass, mmentum and energy equatins. Hwever, analytical slutins are available nly fr very simple situatins, hence mst f the cnvectin heat transfer data is btained thrugh careful experiments, and the equatins suggested fr cnvective heat transfer cefficients are mstly empirical. Since the equatins are f empirical nature, each equatin is applicable t specific cases. Generalizatin has been made pssible t sme extent by using several nn-dimensinal numbers such as Reynlds number, Prandtl number, Nusselt number, Grashff number, Rayleigh number etc. Sme f the mst imprtant and cmmnly used crrelatins are given belw: Heat transfer cefficient inside tubes, ducts etc.: When a fluid flws thrugh a cnduit such as a tube, the fluid flw and heat transfer characteristics at the entrance regin will be different frm the rest f the tube. Flw in the entrance regin is called as develping flw as the bundary layers frm and develp in this regin. The length f the entrance regin depends upn the type f flw, type f surface, type f fluid etc. The regin beynd this entrance regin is knwn as fully develped regin as the bundary layers fill the entire cnduit and the velcity and temperature prfiles remains essentially unchanged. In general, the entrance effects are imprtant nly in shrt tubes and ducts. Crrelatins are available in literature fr bth entrance as well as fully develped regins. In mst f the practical applicatins the flw will be generally fully develped as the lengths used are large. The fllwing are sme imprtant crrelatins applicable t fully develped flws: a) Fully develped laminar flw inside tubes (internal diameter D): Cnstant wall temperature cnditin: Nusselt number, Nu h cd = k f D = 3.66 (7.)
Cnstant wall heat flux cnditin: Nusselt number, Nu h cd = k f D = 4.364 (7.3) b) Fully develped turbulent flw inside tubes (internal diameter D): Dittus-Belter Equatin: h cd Nusselt number, Nu D = = 0.03 Re k f 0.8 D Pr n (7.4) where n = 0.4 fr heating (T w > T f ) and n = 0.3 fr cling (T w < T f ). The Dittus-Belter equatin is valid fr smth tubes f length L, with 0.7 < Pr < 160, Re D > 10000 and (L/D) > 60. Petukhv equatin: This equatin is mre accurate than Dittus-Belter and is applicable t rugh tubes als. It is given by: Nu D Re D Pr f μ = X 8 μ where X = 1.07 + 1.7(Pr b w n / 3 f 1) 8 1/ (7.5) where n = 0.11 fr heating with unifrm wall temperature n = 0.5 fr cling with unifrm wall temperature, and n = 0 fr unifrm wall heat flux r fr gases f in Petukhv equatin is the frictin factr, which needs t be btained using suitable crrelatins fr smth r rugh tubes. μ b and μ w are the dynamic viscsities f the fluid evaluated at bulk fluid temperature and wall temperatures respectively. Petukhv equatin is valid fr the fllwing cnditins: 10 4 < Re D < 5 X 10 6 0.5 < Pr < 00 with 5 percent errr 0.5 < Pr < 000 with 10 percent errr 0.08 < (μ b /μ w ) < 40 c) Laminar flw ver a hrizntal, flat plate (Re x < 5 X 10 5 ): Cnstant wall temperature: h cx Lcal Nusselt number, Nu x = = 0.33 Re k f 0.5 x Pr 1/ 3 (7.6)
Cnstant wall heat flux: h cx 0.5 1/ 3 Lcal Nusselt number, Nu x = = 0.453Re x Pr (7.7) k f The average Nusselt number is btained by integrating lcal Nusselt number frm 0 t L and dividing by L d) Turbulent flw ver hrizntal, flat plate (Re x > 5 X 10 5 ): Cnstant wall temperature: hc L 1/3 0.8 Average Nusselt number, Nu L = = Pr (0.037 ReL - 850) k f (7.8) e) Free cnvectin ver ht, vertical flat plates and cylinders: Cnstant wall temperature: h c L n n Average Nusselt number, Nu L = = c (GrL Pr) = cra L (7.9) k f where c and n are 0.59 and ¼ fr laminar flw (10 4 < Gr L.Pr < 10 9 ) and 0.10 and ⅓ fr turbulent flw (10 9 < Gr L.Pr < 10 13 ) In the abve equatin, Gr L is the average Grashff number given by: 3 gβ (Tw -T ) L Average Grashff Number Gr L = (7.30) υ where g is the acceleratin due t gravity, β is vlumetric cefficient f thermal expansin, T w and T are the plate and the free stream fluid temperatures, respectively and ν is the kinematic viscsity. Cnstant wall heat flux, q W : hx c * 1/5 Lcal Nusselt number, Nu x = = 0.60 (Grx Pr) kf 4 * gβ q w x where Gr x = k f υ The abve equatin is valid fr 10 5 < Gr * x.pr < 10 11 (7.31) f) Free cnvectin ver hrizntal flat plates: h c L Average Nusselt number, Nu L = = c (Gr k f L Pr) n (7.3)
The values f c and n are given in Table 7.3 fr different rientatins and flw regimes. Table 7.3 Values f c and n 10 t 10 Orientatin f plate Range f Gr L Pr c n Flw regime Ht surface facing up r cld 10 5 t X 10 7 0.54 1/4 Laminar surface facing dwn, cnstant T w X 10 7 t 3 X 10 10 0.14 1/3 Turbulent Ht surface facing dwn r cld 3 X 10 5 t 3 X 10 10 0.7 1/4 Laminar surface facing up, cnstant T w Ht surface facing up, cnstant < X 10 8 0.13 1/3 q w 5 X 10 8 t 10 11 0.16 1/3 Ht surface facing dwn, 0.58 1/5 cnstant q w In the abve free cnvectin equatins, the fluid prperties have t be evaluated at a mean temperature defined as T m = T w 0.5(T w -T ). g) Cnvectin heat transfer with phase change: Filmwise cndensatin ver hrizntal tubes f uter diameter D : The heat transfer cefficient fr film-wise cndensatin is given by Nusselt s thery that assumes the vapur t be still and at saturatin temperature. The mean cndensatin heat transfer cefficient, h m is given by: 1/4 3 k f ρ f g h fg h m = 0.75 (7.33) NDμ f ΔT where, subscript f refers t saturated liquid state, N refers t number f tubes abve each ther in a clumn and ΔT = T r T w, T r and T w being refrigerant and utside wall temperatures respectively. Filmwise cndensatin ver a vertical plate f length L: The mean cndensatin heat transfer cefficient, h m is given by, Nucleate pl biling f refrigerants inside a shell: h m 1/4 3 k f ρ f g h fg = 0.943 (7.34) μ f LΔT h r t 3 = C ΔT (7.35) where ΔT is the temperature difference between surface and biling fluid and C is a cnstant that depends n the nature f refrigerant etc.
The crrelatins fr cnvective heat transfer cefficients given abve are nly few examples f sme f the cmmn situatins. A large number f crrelatins are available fr almst all cmmnly encuntered cnvectin prblems. The reader shuld refer t standard text bks n heat transfer fr further details. 7.3. Fundamentals f Mass transfer When a system cntains tw r mre cmpnents whse cncentratin vary frm pint t pint, there is a natural tendency fr mass t be transferred, minimizing the cncentratin differences within the system. The transprt f ne cnstituent frm a regin f higher cncentratin t that f lwer cncentratin is called mass transfer. A cmmn example f mass transfer is drying f a wet surface expsed t unsaturated air. Refrigeratin and air cnditining deal with prcesses that invlve mass transfer. Sme basic laws f mass transfer relevant t refrigeratin and air cnditining are discussed belw. 7.3.1. Fick s Law f Diffusin: This law deals with transfer f mass within a medium due t difference in cncentratin between varius parts f it. This is very similar t Furier s law f heat cnductin as the mass transprt is als by mlecular diffusin prcesses. Accrding t this law, rate f diffusin f cmpnent A m A (kg/s) is prprtinal t the cncentratin gradient and the area f mass transfer, i.e. dc A m A = D ABA (7.36) dx where, D AB is called diffusin cefficient fr cmpnent A thrugh cmpnent B, and it has the units f m /s just like thse f thermal diffusivity α and the kinematic viscsity f fluid ν fr mmentum transfer. 7.3.. Cnvective mass transfer: Mass transfer due t cnvectin invlves transfer f mass between a mving fluid and a surface r between tw relatively immiscible mving fluids. Similar t cnvective heat transfer, this mde f mass transfer depends n the transprt prperties as well as the dynamic characteristics f the flw field. Similar t Newtn s law fr cnvective heat transfer, he cnvective mass transfer equatin can be written as: m = h A Δ (7.37) m c A where h m is the cnvective mass transfer cefficient and Δc A is the difference between the bundary surface cncentratin and the average cncentratin f fluid stream f the diffusing species A. Similar t cnvective heat transfer, cnvective mass transfer cefficient depends n the type f flw, i.e., laminar r turbulent and frced r free. In general the mass transfer cefficient is a functin f the system gemetry, fluid and flw prperties and
the cncentratin difference. Similar t mmentum and heat transfers, cncentratin bundary layers develp whenever mass transfer takes place between a surface and a fluid. This suggests analgies between mass, mmentum and energy transfers. In cnvective mass transfer the nn-dimensinal numbers crrespnding t Prandtl and Nusselt numbers f cnvective heat transfer are called as Schmidt and Sherwd numbers. These are defined as: h ml Sherwd number, Sh L = (7.38) D ν Schmidt number, Sc = (7.39) D where h m is the cnvective mass transfer cefficient, D is the diffusivity and ν is the kinematic viscsity. The general cnvective mass transfer crrelatins relate the Sherwd number t Reynlds and Schmidt number. 7.4. Analgy between heat, mass and mmentum transfer 7.4.1. Reynlds and Clburn Analgies The bundary layer equatins fr mmentum fr a flat plate are exactly same as thse fr energy equatin if Prandtl number, Pr = 1, pressure gradient is zer and viscus dissipatin is negligible, there are n heat surces and fr similar bundary cnditins. Hence, the slutin fr nn-dimensinal velcity and temperature are als same. It can be shwn that fr such a case, Nu h f Re.Pr ρvc c Stantn number, St = = = p (7.40) where f is the frictin factr and St is Stantn Number. The abve equatin, which relates heat and mmentum transfers is knwn as Reynlds analgy. T accunt fr the variatin in Prandtl number in the range f 0.6 t 50, the Reynlds analgy is mdified resulting in Clburn analgy, which is stated as fllws. / 3 f St.Pr = (7.41) 7.4.. Analgy between heat, mass and mmentum transfer The rle that thermal diffusivity plays in the energy equatin is played by diffusivity D in the mass transfer equatin. Therefre, the analgy between mmentum and mass transfer fr a flat plate will yield: Sh h ml ν D = = Re.Sc D VL ν h m V f = (7.4)
T accunt fr values f Schmidt number different frm ne, fllwing crrelatin is intrduced, Sh 3 f Sc / = (7.43) Re.Sc Cmparing the equatins relating heat and mmentum transfer with heat and mass transfer, it can be shwn that, / 3 h c α = (7.44) c ph ρ m D This analgy is fllwed in mst f the chemical engineering literature and α/d is referred t as Lewis number. In air-cnditining calculatins, fr cnvenience Lewis number is defined as: Lewis number, Le / 3 α = (7.45) D The abve analgies are very useful as by applying them it is pssible t find heat transfer cefficient if frictin factr is knwn and mass transfer cefficient can be calculated frm the knwledge f heat transfer cefficient. 7.5. Multimde heat transfer In mst f the practical heat transfer prblems heat transfer ccurs due t mre than ne mechanism. Using the cncept f thermal resistance develped earlier, it is pssible t analyze steady state, multimde heat transfer prblems in a simple manner, similar t electrical netwrks. An example f this is transfer f heat frm utside t the interirs f an air cnditined space. Nrmally, the walls f the air cnditined rms are made up f different layers having different heat transfer prperties. Once again the cncept f thermal resistance is useful in analyzing the heat transfer thrugh multilayered walls. The example given belw illustrates these principles. Multimde heat transfer thrugh a building wall: The schematic f a multimde heat transfer building wall is shwn in Fig. 7.3. Frm the figure it can be seen that: (T1-T ) Q = (7.46a) R 1- R = R R + ( R +R +R ) + R R cnv, rad, cnv,1 rad,1 ttal w,3 w, w,1 R cnv, +R rad, R cnv,1+rrad,1 ttal ( ) ( ) ( ) ttal w 1 (7.46b) R = R + R + R (7.46c) Q 1- =UA(T-T 1 ) (7.46d)
1 where, verall heat transfer cefficient, U = R A ttal q rad q rad T 1 T T 1 Rm Rm 1 q T cnv T 1 T q cnv 3 1 R rad, R rad,1 T R w,3 R w, R w,1 T 1 R cnv, R cnv,1 T R R w R 1 T 1 Cmpsite cylinders: Fig. 7.3. Schematic f a multimde heat transfer building wall The cncept f resistance netwrks is als useful in slving prblems invlving cmpsite cylinders. A cmmn example f this is steady state heat transfer thrugh an insulated pipe with a fluid flwing inside. Since it is nt pssible t perfectly insulate the pipe, heat transfer takes place between the surrundings and the inner fluid when they are at different temperatures. Fr such cases the heat transfer rate is given by: Q = U A (T T ) (7.47) i
where A is the uter surface area f the cmpsite cylinder and U is the verall heat transfer cefficient with respect t the uter area given by: 1 U A 1 = h A i i ln(r/r + πlk 1 m ) ln(r3/r) 1 + + πlk h A in (7.48) In the abve equatin, h i and h are the inner and uter cnvective heat transfer cefficients, A i and A are the inner and uter surface areas f the cmpsite cylinder, k m and k in are the thermal cnductivity f tube wall and insulatin, L is the length f the cylinder, r 1, r and r 3 are the inner and uter radii f the tube and uter radius f the insulatin respectively. Additinal heat transfer resistance has t be added if there is any scale frmatin n the tube wall surface due t fuling. T, h Insulatin Fluid in T i, h i Fluid ut Tube wall Fig. 7.4. Cmpsite cylindrical tube 7.6. Heat exchangers: A heat exchanger is a device in which heat is transferred frm ne fluid stream t anther acrss a slid surface. Thus a typical heat exchanger invlves bth cnductin and cnvectin heat transfers. A wide variety f heat exchangers are extensively used in refrigeratin and air cnditining. In mst f the cases the heat exchangers perate in a steady state, hence the cncept f thermal resistance and verall heat transfer cefficients can be used very cnveniently. In general, the temperatures f the fluid streams may vary alng the length f the heat exchanger. T take care f the temperature variatin, the cncept f Lg Mean Temperature Difference (LMTD) is intrduced in the design f heat exchangers. It is defined as: ΔT1 ΔT LMTD = (7.49) ln ( ΔT / ΔT ) where ΔT 1 and ΔT are the temperature difference between the ht and cld fluid streams at tw inlet and utlet f the heat exchangers. 1
If we assume that the verall heat transfer cefficient des nt vary alng the length, and specific heats f the fluids remain cnstant, then the heat transfer rate is given by: Q = U als Q = U i A (LMTD) = U A A (LMTD) = U i i A i ΔT ln ( ΔT ΔT 1 1 ln ( ΔT ΔT / ΔT ΔT 1 1 / ΔT ) ) (7.50) the abve equatin is valid fr bth parallel flw (bth the fluids flw in the same directin) r cunterflw (fluids flw in ppsite directins) type heat exchangers. Fr ther types such as crss-flw, the equatin is mdified by including a multiplying factr. The design aspects f heat exchangers used in refrigeratin and air cnditining will be discussed in later chapters. Questins: 1. Obtain an analytical expressin fr temperature distributin fr a plane wall having unifrm surface temperatures f T 1 and T at x 1 and x respectively. It may be mentined that the thermal cnductivity k = k 0 (1+bT), where b is a cnstant. (Slutin). A cld strage rm has walls made f 0.3 m f brick n utside fllwed by 0.1 m f plastic fam and a final layer f 5 cm f wd. The thermal cnductivities f brick, fam and wd are 1, 0.0 and 0. W/mK respectively. The internal and external heat transfer cefficients are 40 and 0 W/m K. The utside and inside temperatures are 40 0 C and -10 0 C. Determine the rate f cling required t maintain the temperature f the rm at -10 0 C and the temperature f the inside surface f the brick given that the ttal wall area is 100 m. (Slutin) 3. A steel pipe f negligible thickness and having a diameter f 0 cm has ht air at 100 0 C flwing thrugh it. The pipe is cvered with tw layers f insulating materials each having a thickness f 10 cm and having thermal cnductivities f 0. W/mK and 0.4 W/mK. The inside and utside heat transfer cefficients are 100 and 50 W/m K respectively. The atmsphere is at 35 0 C. Calculate the rate f heat lss frm a 100 m lng pipe. (Slutin) 4. Water flws inside a pipe having a diameter f 10 cm with a velcity f 1 m/s. the pipe is 5 m lng. Calculate the heat transfer cefficient if the mean water temperature is at 40 0 C and the wall is isthermal at 80 0 C. (Slutin) 5. A lng rd having a diameter f 30 mm is t be heated frm 400 0 C t 600 0 C. The material f the rd has a density f 8000 kg/m 3 and specific heat f 400 J/kgK. It is placed cncentrically inside a lng cylindrical furnace having an internal diameter f 150 mm. The inner side f the furnace is at a temperature f 1100 0 C and has an
emissivity f 0.7. If the surface f the rd has an emissivity f 0.5, find the time required t heat the rd. (Slutin) 6. Air flws ver a flat plate f length 0.3 m at a cnstant temperature. The velcity f air at a distance far ff frm the surface f the plate is 50 m/s. Calculate the average heat transfer cefficient frm the surface cnsidering separate laminar and turbulent sectins and cmpare it with the result btained by assuming fully turbulent flw. (Slutin) Nte: The lcal Nusselt number fr laminar and turbulent flws is given by: 1/ 1/3 laminar : Nu = 0.331Re Pr x x 0.8 1/3 x Pr turbulent: Nu x = 0.088Re 5 Transitin ccurs at Re x.trans = X 10. The frced cnvectin bundary layer flw begins as laminar and then becmes turbulent. Take the prperties f air t 3-5 be ρ = 1.1 kg/m, μ = 1.7 X 10 kg/m s, k = 0.03 W/mK and Pr = 0.7. 7. A vertical tube having a diameter f 80 mm and 1.5 m in length has a surface temperature f 80 0 C. Water flws inside the tube while saturated steam at bar cndenses utside. Calculate the heat transfer cefficient. (Slutin) 0 Nte: Prperties f saturated steam at bar: T sat = 10. C, h fg = 0 kj/kgk, 3 ρ = 1.19 kg/m ; Fr liquid phase at 100 0 3 C: ρ = 958 kg/m, c = 419 J/kgK, -3 μ L = 0.79X10 kg/m s and Pr = 1.73. 8. Air at 300 K and at atmspheric pressure flws at a mean velcity f 50 m/s ver a flat plate 1 m lng. Assuming the cncentratin f vapur in air t be negligible, calculate the mass transfer cefficient f water vapur frm the plate int the air. The diffusin f water vapur int air is 0.5 X 10-4 m /s. The Clburn j-factr fr heat transfer cefficient is given by j H =0.096 Re -0.. (Slutin) 9. An il cler has t cl il flwing at 0 kg/min frm 100 0 C t 50 0 C. The specific heat f the il is 000 J/kg K. Water with similar flw rate at an ambient temperature f 35 0 C is used t cl the il. Shuld we use a parallel flw r a cunter flw heat exchanger? Calculate the surface area f the heat exchanger if the external heat transfer cefficient is 100 W/m K. (Slutin) L p