EEN-E3003, Industrial drying and evaporation processes Equations, Mollier diagram, Thermodynamic properites of humid air, h,s diagram of water

Transcription

1 EEN-E3003, Industrial drying and evaporation processes Equations, Mollier diagra, Therodynaic properites of huid air, h,s diagra of water Therodynaics of huid air Air huidity [kg/kg da ] v x =, (1) da Total pressure [Pa] v = ass of water vapor [kg] da = ass of dry air [kg] ptot = pda + pv, (2) pda = partial pressure of dry air [Pa] pv = vapor pressure [Pa] The dependence beween the air huidity and partial vapor pressure p p - p v x = 0.622, (3) tot v pv = vapor pressure [Pa] ptot = total pressure [Pa] Figure is the ratio of olar asses of water vapor and dry air (Mw/Mda = 0.018/0.0289) An approxiation equation for vapor pressure [Pa] 11,78 (T - 372,79) v ' = p exp( ), (4) T - 43,15 p 0 T = teperature [K] p0 = 10 5 Pa

2 Relative huidity pv(t) j =, (5) p '(T) v pv (T) = vapor pressure at the teperature of T [Pa] pv (T) = saturated vapor pressure at the teperature of T [Pa] Enthalpy of huid air [kj/kg da ] h = hda + xhv, (6) hda = enthalpy of dry air [kj/kgda] x = air huidity [kg/kgda] hv = enthalpy of vapor [kj/kg] The enthalpy of huid air when average specific heat capacities are used [kj/kg da ] h = cpdat + x(cpvt ), (7) cpda = specific heat capacity of dry air [kj/(kg o C)] t = teperature [ o C] x = air huidity [kg/kgda] cpv = specific heat capacity of water vapor [kj/(kg o C)] An approxiation equation for vaporization heat of water [J/kg] lv = t (8) lv = vaporization heat [J/kg] t = teperature [ o C]

3 Definitions of oisture and dry solids contents Moisture content, wet basis w = w + w ds, (9) w = oisture content on wet basis [kg/kgtot] w = ass of water [kg] ds = ass of dry solids [kg] Moisture content, dry basis u = w, (10) ds u = oisture content on dry basis [kg/kgds] w = ass of water [kg] ds = ass of dry solids [kg] Dry solids content C + ds =, (11) ds w C = dry solids content [kgka/kgtot] w = ass of water [kg] ds = ass of dry solids [kg] The dependence between oisture contents calcualted on dry and wet basis u w = 1+ u (12) w u = 1- w. (13) u = oisture content, dry basis [kg/kgds] w = oisture content, wet basis [kg/kgtot]

4 Diffusion Diffusion flux of a coponent A [ol/( 2 s)] j A ca = -DAB, (14) Z DAB = diffusion coefficient [ 2 /s] ca = concentration of a coponent A [ol/ 3 ] Z = distance [] Net flux of a coponent A [ol/( 2 s)] J A ca = vc A - DAB, (15) Z v = average speed of particles [/s] DAB = diffusion coefficient/diffusivity [ 2 /s] ca = concentration of a coponent A [ol/ 3 ] Z = distance [] Fuller odel for diffusion coefficient [ 2 /s] D AB 1,75 T (1/M + 1/M ) = , (16) p[(σv + Σv A 1/3 A) 1/2 B 1/3 2 B) ] T = teperature [K] MA = olar ass of a coponent A [kg/ol] MB = olar ass of a coponent B [kg/ol] p = total pressure [Pa] SnA = atoic diffusion volue of water vapor = 12.7 SnB = atoic diffusion volue of dry air = 20.1 The effective diffusivity of water vapor D eff in a porous aterial [ 2 /s] D F =, (17) ψ eff D AB DAB = diffusion coefficient of water vapor in the air [ 2 /s]

5 F = porosity of aterial y = tortuosity The effective diffusivity of water vapor in porous aterial when the Knudsen diffusion coefficient is considered [ 2 /s] ( D ) D + ( D ) 1 eff F = AB k, (18) Ł ψ ł DAB = diffusion coefficient of water vapor in the air [ 2 /s] F = porosity of aterial y = tortuosity Dk = the Knudsen diffusion coefficient [ 2 /s] Capillary action Capillary pressure Dp c [Pa] Capillary rise 2γ Δp C = cos q, (19) R g = water tension [N/] R = radius of pore [] q = contact angle [ o ] 2g cosq Z = rgr, (20) Z = height of the water colun [] g = surface tension [N/] R = radius of pore [] q = contact angle [ o ] g = gravitational constant [9,81/s 2 ] r = density of liquid [kg/ 3 ]

6 The influence of capillary pressure on saturated vapor pressure [Pa] p v ' v 2γvM - cosθ rrt = p e, (21) pv = saturated vapor pressure on a flat surface [Pa] g = surface tension [N/] r = radius of a pore [] q = contact angle [ o ] R = gas constant [= J/(olK)] T = teperature [K] Penetration depth of liquid in porous aterial [] z = 2R g cosθ + (p 4η s - p o )R 2 t, (22) z = penetration depth [] g = surface tension [N/] R = radius of pore [] q = contact angle [ o ] ps = pressure of liquid in porous aterial [Pa] po = pressure of gas in porous aterial [Pa] h = dynaic viscocity [kg/s] t = tie [s] Calculating drying and evaporation rates Evaporation rate of water through a boundary layer fro a water surface [kg/( 2 s)] '' p p - p o o v = M H2O k cln, (23) RT p o - p v' ( t p) = evaporation rate [kg/( 2 s)] MH2O = olar ass of water 0.018kg/ol po = total pressure [Pa] R = gas constant = J/(olK) T = teperature of the boundary layer [K] kc = ass transfer coefficient [/s] pv = vapor pressure in the surroundings [Pa] pv (tp) = vapor pressure on the surface [Pa]

7 Energy balance of a oist particle that evaporates (unifor oisture and teperature distribution inside the particle is assued) dt cp = -''lva - α(t - t 0 )A, (24) dτ = ass of the particle (dry or oist) [kg] cp = specific heat capacity of the particle per dry of oist ass [J/(kg o C)] t = teperature [ o C] t = teperature of the particle [ o C] t0 = teperature of the surroundings [ o C] t = tie [s] = evaporation rate [kg/( 2 s)] lv = vaporization heat at t [J/kg] A = evaporation surface [ 2 ] a = heat transfer coefficient [W/ 2 K] Specific heat capacity of a oist particle [J(kgK)] cp = cpds + cpwu, (25) cpds = specific heat capacity of dry solids [J/(kg o C)] cpw = cpecific heat capacity of water [J/(kg o C)] u = oisture content on dry basis [kg/kgds] The analogy between heat and ass transfer k c α ρc 1-n = Le, (26) p kc = ass transfer coefficient [/s] a = heat transfer coefficient [W/( 2 K)] Le = Lewisi Nuebr = Pr/Sc rcp = rdacpda + rvcpv [J/( 3o C)], where rda = density of dry air at the teperature of boundary layer [kg/ 3 ] cpda = specific heat capacity of dry air at the teperature of boundary layer [J/(kg o C)] rv = density of vapor at the teperature of boundary layer [kg/ 3 ] cpv = specific heat capacity of vapor at the teperature of boundary layer [J/(kg o C)]

8 Diensioless nubers (drying/evaporation) Nusselt nuber : Reynolds nuber Prandtl nuber Sherwood nuber Schidt nuber a Nu = L l vl Re = ν (27a) (27b) μc p Pr = (27c) λ k cl Sh = D AB D AB (27d) n Sc = (27e) Lewis nuber Le = Pr/Sc, (27f) a = heat transfer coefficient [W/( 2 K)] L = characteristic length [] l = heat conductivity of vapor + dry air ixture in the boundary layer [W/] n = kineatic viscosity of vapor + dry air ixture in the boundary layer [ 2 /s] = dynaic viscosity of vapor + dry air ixture in the boundary layer [kg/s] cp = specific heat capacity of vapor + dry air ixture [kj/kg o C] DAB = diffusion coefficient of vapor + dry air ixture in the boundary layer [ 2 /s] kc = ass transfer coefficient [/s] Drying rate in the periof constant drying rate [kg/s] ev ( t o - t ) l ( t ) & = Aα, (28) v & ev = evaporation rate [kg/s] A = evaporation surface [ 2 ] a = heat transfer coefficient [W/( 2 K)] to = teperature of the surroundings (usually the air) [ o C] t = teperature of the particle [ o C] lv(t) = vaporization heat at the teperature of the particle [ o C]

9 Evaporation rate in the period of constant drying rate, when the state of drying gas changes [kg/s] v α ( t o1 - t )- ( t o2 - t ) ( t ) ( t o1 - t ) ln ( t - t ) & ev = A, (29) l o2 & ev = evaporation rate [kg/s] A = evaporation surface [ 2 ] a = heat transfer coefficient [W/( 2 K)] to1 = teperature of the drying gas at inlet [ o C] ti2 = teperature of the drying gas at outlet [ o C] t = teperature of the aterial/particles [ o C] lv(t) = vaporization heat at the teperature of the aterial/particles [ o C] Darcy s law [kg/( 2 s)] K p '' = -, (30) ν z = flux/ass flow rate of the fluid [kg/( 2 s)] K = pereability [ 2 ] u = kineatic viscosity [ 2 /s] p = pressure [Pa] z = distance [] Characteristic drying curve Drying tie for a plate when the charqacteristic drying curve is used [s] ut u cr Zρ dsu cr d(u/ucr ) τ = -, (31a) 2'' f(u/u ) o uo u cr cr t = drying tie [s] L = thickness of the plate [] rds = density of dry solids [kgds/ 3 ] ucr = critical oisture content [kg/kgds] o = evaporation rate in the period of constant drying rate [kg/ 2 s] uo = oisture content in the beginning of drying [kg/kgds] ut = oisture content in the end of drying [kg/kgds]

10 f(u/ucr) = characteristic drying curve Drying tie calculated using the characteristic drying curve when the drying rate in the period of constant drying is reported per ass of the particle/bed [s] u τ = - ut ucr cr ' o uo ucr d(u/u f(u/u cr cr ) ) (31b) t = drying tie [s] ucr = critical oisture content [kg/kgds] o = drying rate in the period of constant drying per ass of the particle/bed [kgh2o/(kgdss)] ut = oisture content in the end of drying [kg/kgds] f(u/ucr) = characteristic drying curve Drying tie calculated using the characteristic drying curve when the drying rate in the period of constant drying is reported per volue of the particle/bed [s] ρdsu τ = - ''' o cr ut u cr uo u cr d(u/u f(u/u cr cr ) ) (31c) t = drying tie [s] rka = density of dry solids of the particle/bed [kgds/ 3 ] ucr = kriittinen kosteussuhde [kg/kgka] ''' o = drying rate in the period of constant drying per volue of the particle/bed [kgh2o/(kgdss)] ut = oisture content in the end of drying [kg/kgds] f(u/ucr) = characteristic drying curve

11 Energy and ass balance of a dryer Mass balance over the drying chaber, when drying gas is air/other gas & ds (uin uout) = & da (xout xin), (32) & & ds da = ass flow rate of dry solids [kgds/s] = ass flow rate of dry air (drying gas) [kgda/s] uin = initial oisture content of the aterial [kg/kgds] uout = final oisture content of the aterial [kg/kgds] x out = absolute huidity of air (drying gas) after the drying chaber [kg/kg da] x in = absolute huidity of air (drying gas) after the drying chaber [kg/kg da] Energy balance over the drying chaber, when drying gas is air/other gas & ki (hin-hout) + & ds ( iin iout) + F + P - Q = 0, (33) & & ds da = ass flow rate of dry solids [kgds/s] = ass flow rate of dry air (drying gas) [kgda/s] hin = enthalpy of drying air/gas before the drying chaber [kj/kgda] hout = enthalpy of drying air/gas after the drying chaber [kj/kgda] iin = enthalpy of aterial before the drying chaber [kj/kgds] iout = enthalpy of aterial after the drying chaber [kj/kgds] F = excess heat into the drying chaber [kw] P = echanical work input into teh drying chaber [kw] Q = heat losses of the drying chaber [kw] Enthalpy of huid air in the energy balance [kj/kg da ] h = cpdat + x(cpvt ), (34) cpda = specific heat capacity of dry air [kj/(kg o C)] t = air teperature [ o C] x = absolute huidity of air [kg/kgda] cpv = specific heat capacity of vapor [kj/(kg o C)]

12 Entahlpy of oist aterial in the energy balance [kj/kg ds ] i = cpdst + ucpwt (35) cpds = specific heat capacity of dry solids [kj/kg o C] cpw = specific heat capacity of liquid water [kj/kg o C] u = oisture content of aterial [kg/kgds] t = teperature of the aterial [ o C] Eelctricity consuption of a fan Dp P = V & (36) h h s V & = Volue flow rate of drying gas just before the fan [ 3 /s] Dp = pressure difference over the fan [Pa] h = echanical efficiency of the fan [-] hs = isentropic efficiency of the fan [-] When superheated stea is used as a drying gas, enthalpy values for stae/condensate/water are taken fro stea tables. The enthalpy of aterial is calculated using Eg, (35) Desing of a dryer Length of the dryer for a continuous belt dryer & dsτ L = ρ ZW ds (37) L = lengthg of the dryer [] & ds = ass flow rate of dry solids [kgds/s ] t = residence tie of aterial in the dryer [s] Z = bed height [] W = width of the conveyor []

13 Drying of wood-based bioass Lower heating value (LHV) of oist bioass/fuel per vet basis [MJ/kg tot ] qwet,w = qi(1- w) 2.443w (38) qwet,w = LHV of oist bioass [MJ/kgtot] qi = LHV of copletely dry bioass/fuel [MJ/kgds] w = oisture content of bioass [% wb] Lower heating value (LHV) of oist bioass/fuel per dry basis [MJ/kg ds ] Fuel input [W] qwet,u = qi 2.443u (39) qwet,u= LHV of oist bioass/fuel [MJ/kgds] qi = LHV of copletely dry bioass/fuel [MJ/kgds] u = oisture content of bioass/fuel [% db] Φ fuel = & q (40) fuel fuel & fuel = ass flow rate of fuel [kgtot/s or kgds/s] qfuel = LHV of fuel [J/kgtot or J/kgds] Boiler efficiency Q Φ η - in loss b = = 1 (41) Φ fuel Φ fuel Qin = heat input fro the boiler into the cycle process [W] Ffuel = fuel input [W] Floss = heat losses fro the boiler into the surroundings [W]

14 Power-to-heat ratio of a CHP plant P a = Q (42) P = electricity production [W] Q = heat production [W] Efficiency of a CHP plant η CHP P + Q = (43) Φ fuel Ffuel = fuel input [W] P = electricity production [W] Q = heat production [W] Efficiency of a cycle process in a condensing power plant P Q t out h th = = 1-, (44) Qin Q in Pt = shaft power of a turbine [W] Qin = heat input fro the boiler into the cycle process [W] Qout = heat output fro the cycle process into the surroundings [W] Efficiency of a condensing power plant P η e = (45) Φ fuel P = electricity production after the generator [W] Fpa = fuel input [W]

15 The change of boiler losses as a result of drying when the outlet teperature of flue gases, the excess air factor and other heat losses reain the sae as in the case of oist bioass/fuel [W] loss2 loss1 fuel ( u 2 u 1 ) h fg Φ - Φ = & - (46) Floss2 = boiler losses for dry fuel [W] Floss1 = boiler losses for oist fuel [W] & fuel = ass flow rate of dry bioass/fuel [kgds/s] u2 = oisture content after drying [% d.b.] u1 = oisture content before drying [% w.b.] hsk = the outgoing enthalpy of flue gases [J/kg] Consuption of arginal fuel in a condensing power plant that cobusts bioass [W] P & & - q bio bio ar = (47) η thηbηgq ar q ar & ar = ass flow rate of arginal fuel [kg/s] hth = efficiency of cycle process [-] hb = boiler efficiency hg = efficiency of the generator qar = LHV of arginal fuel [J/kg] & = ass flow rate of bioass (oist or dry) [kg/s] bio qbio = LHV of bioass per kilogra of oist or dry fuel [MJ/kg] Consuption of arginal fuel in a CHP plant that cobusts bioass [W] Q + αq & & - q heat heat bio bio ar = (48) ηchpq ar q ar & ar = ass flow rate of bioass [kg/s] Qheat = heat production [W] a = power-to-heat ratio hchp = efficiency of the CHP plant (see Eq. (43)) & bio = ass flow rate of bioass (oist or dry) [kg/s] qbio = LHV of bioass per kilogra of oist or dry fuel [J/kg] qar = LHV of arginal fuel [J/kg]

16 Evaporators Dry solids content C = ds ds + w (49) ds = ass of dry solids in solution [kg] w = ass of water [kg] Mass flow rate of water evaporated in an evaporator [kg/s] & w = & feed C 1- Ł C 1 2 ł (50) & feed = ass flow rate of feed (incoing solution) [kg/s] C1 = dry solids content of feed [kgds/kgtot] C2 = dry solids content of concentrate [kgka/kgtot] Heat transfer rate through the evaporation surface [W] F = kadt (51) k = heat transfer coefficient [W/( 2 K)] A = heat transfer/evaporation surface [ 2 ] Dt = teperature difference between the heat source and solution [ o C] Heat transfer coefficient k for a plate [W/( 2 K)] 1 k = 1 α + s λ + δ λ s d α 2 (52) k = heat transfer coefficient [W/( 2 K)] a1 = convective heat transfer coefficient on the solution side [W/( 2 K)] a2 = convective heat transfer coefficient on side of heat source [W/( 2 K)] s = wall thickness [] ls = heat conductivity of the wall [W/(K)]

17 d = thickness of fouling layer [] l d = heat conductivity of the fouling layer [W/(K)] Electricity deand of a copressor [W] P = & Δh η (53a) Δh η is Δh =, (53b) is & = ass flow rate of gas through the copressor [kg/s] Dh = real enthalpy change of the gas over the copressor [J/kg] Dhis = change of enthalpy in isentropic copression [J/kg] his = isentropic efficiency h = echanical efficiency Enthalpies of water/stea/condensate fro the stea table can be used when energy balances are ade for evaporators if anything else is not entioned. The dependence between teperature and pressure for an ideal gas in an isentropic copression T 2 Ł T 1 = ł Ł p p 2 1 ł R Mcp (54) T2 = teperature after copression [K] T1 = teperature before copression [K] p2 = pressure after copression [Pa] p1 = pressure before copression [Pa] R = gas constant [8.314 J/(olK)] M = olar ass of gas [kg/ol] cp = average heat capacity of the gas over the copression process [J/kgK]

18 Boiling point elevation [ o C,K] Derived based on the equilibriu theory 2 RT ΔT = - (55) l v K ( T ) lna K DT = boiling poiont elevation [K] R = gas consatnt [8.314 J/(olK)] TK = boiling point of a pure water [K] lv = waporization heat of water at TK [J/ol] a = activity of water in solution Activity when aounts of substances of species are used n a = γ n w tot (56) g = activity coefficient [-] nw = ole aount of water in solution [ol] ntot = ole aount of the whole solution [ol] An experietal equation ΔTBPE = Kb * i * (57) Kb = constant [J/K] i = van t Hoff coefficient = olality [ol/kg] NPSH-value (Net Positive Suction Head) [] NPSH = Hstat + pi / (ρ g) ploss/ (ρ g) pn /(ρ g) (58) Hstat = static suction height [] pi = absolute pressure in a suction tank [Pa] Hpipe = pressure loss in a suction pipeline [] pn = vapor pressure at puping teperature [Pa] ρ = density of liquid [kg/ 3 ] g = gravity = /s 2 Hstat has a negative sign if the pup is above the water surface and a positive sign if it is situated below the water surface.

19 Average velocity of substance in a pipe [/s] v = 4V& 2 pd (59) V & = volue flow rate [ 3 /s] D = diaeter of a pipe [] Mass flow rate of leaking air & leaking Dp = V Dτ (60) Dp/Dt = pressure increase in a vessel with vacuu [bar/in] V = volue of the vessel [ 3 ] Other equations the ideal gas equation of state Consentration PV = nrt (61) p = pressure [Pa] V = volue [ 3 ] n = aount of substance [ol] R = gas constant [8.314 J/olK] T = teperature [K] c = n V (62) n = aount of substance [ol] V = volue [ 3 ] Molality n M = (63) n = aount of substance [ol] = ass [kg]

20 Diagras and tables on next pages

21 Mollier diagra of huid air

22 Kostean ilan terodynaaiset oinaisuudet

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24 Specific heat capacities of dry air and vapor as function of teperature T [K] t [ C] c p,dry air [kj/kg C] c p, vapor [kj/kg C]

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