Model Equation For Heat Transfer Coefficient Of Air In A Batch Dryer

International Journal of Scientific & Engineering Research, Volume 5, Issue 11, November-2014ISSN 2229-5518121Model Equation for Heat Transfer Coefficient ofAir in a Batch DryerEmenike N. Wami, Moses Onuigezhe IbrahimAbstract— Heat transfer coefficients of dryers are useful tools for correlation formulation and performance evaluation of process design ofdryers as well as derivation of analytical model for predicting drying rates. A model equation for predicting heat transfer coefficient of air ina batch dryer using BuckingHam Pi-theorem and dimensional analysis at various air velocities has been formulated. The model wasvalidated by drying unripe plantain chips in a batch dryer at air velocities between 0.66 and 1.20m/s at corresponding temperaturesbetween 42 and 66oC. Based on the analogy of heat and mass transfer rate equations for constant drying period, the prediction from thedeveloped model agreed reasonably with the experimental data.Index Terms—batch dryer, Buckinham pi-theorem, drying rate, heat transfer coefficient, model equation—————————— ——————————1 INTRODUCTIONDrying is a kinetic process that involves the removal ofliquid, usually water from a moist material: solid, liquidor gas. The use of heat to remove liquid distinguishesdrying from mechanical methods of removing water,such as: centrifugation, decantation, sedimentation andfiltration in which no change in phase from liquid to vapour is experienced [1],[2].The study of convective heat transfer is centered onways and means of determining the heat transfer coefficient, h for various flow regions (laminar, transition orturbulent flow) and over various geometries and configurations. The local and average heat transfer coefficientmay be correlated by (3) and (4) respectively:IJSERThe application of heat to remove moisture is widelyused in the food industry to reduce moisture contents tolevels considered safe for storage in order to prolong thelife span of the food item [3],[4],[5],[6],[7],[8]. Also highmoisture contents of moist food and agricultural materials constitute additional cost in bulk handling and transportation and must be removed in a manner that guarantees product quality [9],[10]. In the drying of solids toremove water a specialized device called dryer is used,and the desirable end products are in solid form. Thefinal moisture contents of the dried solids are usually lessthan1%. The chemistry of drying a moist material can berepresented as:(1)Moist material Heatsolid vapourWhen heat transfer by pure convection is used to dry awet solid, the heat supplied is solely by sensible heat inthe drying gas stream. A dynamic equilibrium exist between the rate of heat transfer to the material and therate of vapour (mass) removal from the surface at instance, (that is, drying rate) and may be represented asfollows:dxdt ℎ𝐴 ΔTλ(2)The area of the heat and mass transfer may be assumedto be approximately equal — Emenike N. Wami is a Professor of Chemical Engineering in Rivers StateUniversity of Science and Tecxhnology,Nigeria, PH- 2348033131062. Email: wami.emenike @ust.edu.ng Moses Onuigezhe Ibrahim is currently pursuing a Masters degree programin Chemical engineering in Rivers State University of Science andTecxhnology,NigeriaIJSER 2014http://www.ijser.orgNux f (x , Rex , Pr )(3)Nux f (Rex , Pr )(4)where the subscript x emphasize the condition at aparticular location on the surface.The problem of convection involves how these functions are obtained, there are two approaches: theoreticaland experimental. Theoretical approach involves solvingthe boundary layer equation for a particular geometryand equation such as (5)hL T Nu K y 0(5) ywhich is a dimensionless temperature gradient at the surface.In the experimental approach, for a prescribed geometry in a parallel flow, if heated, convection heat transfercoefficient which is an average associated with the entiresystem could then be computed from Newton’s law ofcooling. And from the knowledge of the characteristiclength and the fluid properties, the Nusselt, Reynoldsand Prandtl numbers could be computed from their definitions.Meanwhile, the relevant dimensionless parameters for1.2low-speed, forced convection boundary layer have beenobtained by non-dimensionalizing the differential equation that describes the physical process occurring withinthe boundary layer. An alternative approach is the use ofdimensional analysis in the form of Buckingham Pi theorem. The success of the theorem depends on the ability toselect from intuition the various parameters that influ-

International Journal of Scientific & Engineering Research, Volume 5, Issue 11, November-2014ISSN 2229-5518122ence the problem. Therefore, knowing before hand thath f(K, CP , ρ, µ, V, L)(6)One could use the Buckingham Pi theorem to obtain h, in(6) [12].2 MODELLING AND EXPERIMENTALVALIDATION2.1 Model FormulationEtebu and Josiah [13] suggested that in order to successfully create non-dimensional groups, each time a needarises, a set of rules must be followed; the Raleigh method and Buckingham’s Pi-Theorems are reliable.The Raleigh method is an elementary technique for finding a functional relationship between variables. Althoughvery simple, the method does not provide any information concerning the number of dimensionless groupthat can be obtained. Another drawback of the method isthat it can only be used for the determination of the expression for variables that depend on a maximum ofthree or four independent variables.In the determination of heat transfer coefficient therefore,it is necessary to note thatfα(h, K V , D, Re, Pr) 0Re Pr �KgmML-33Velocitym/sLT-14Viscosity (absolute)5Thermal Conductivity6Heat CapacityV𝜇KV7Heat Transfer Coeffi-CPHDimensionL-3kgm-1s-1ML-1T-1kgms ing up and solving equations for π groups, usingthe dimensions in Table 1, we obtain(9)f { α (h, KV , D, ρ, V, μ, CP )} 0(10)where α is a constant.Choosing M, L, T, and K as fundamental dimension formass, length, time and temperature, implies that in (10),the number of fundamental dimension, m, is 4, while thenumber of quantities, n 7 as shown in Table 1Therefore number of π groups is, N𝛑 7 – 4 3Since m is 4, there will be four repeating quantities: Geometric property (D). Flow property (V), fluid property (ρ) and heat property (C P). A 𝛑 group is a function of therepeating variables and one of the remaining variables.Thus the 3π-terms as functions of the repeating variablesare as follows:(14)K𝑣Or π1 (15)ρVDCPUsing similar procedures for (12) and (13), we obtain values for π2 and π3 as in (16) and (17) respectivelyµ(16)ρVDπ3 (8)(11)(12)(13)π1 D 1 V 1 ρ 1 CP 1 Kπ2 Therefore𝛑-termsπ1 Dw1 Vx1 ρy1 CP Z1 K 𝑣π2 Dw2 Vx2 ρy2 CP Z2 µπ3 Dw3 V x3 ρy3 CP Z3 hSIJSERThe BuckingHam’s pi-theorem is an improvement overthe Raleigh’s method. Apart from its advantage of beingable to handle large sets of variables, it gives a ready clueon how many dimensionless groups are designated by Pi.Where,TABLE 1DIMENSION OF QUANTITIESh(17)ρVCPCombining (15), (16) and (17), we havekvµhf α ρVDC,, 0(18)P ρVD ρVCPHencehρVCP α µ,kvρVD ρVDCP(19) From (15), (16) and (17)π3 α(π2 , π1 )(20)Substituting the numerical values of the quantities in (19)from Table 3, the numerical values of π1 , π2 and π3 areobtained as shown in Table 4.Plotting the (π3 ) against (π1 ) , (π2 ), in (20), we obtaingraphs which when regressed yields the theoretical heattransfer coefficient (hB ).The regression gives the equations as:π3 α{6.37(π1 ) 0.029}IJSER 2014http://www.ijser.org(21)

International Journal of Scientific & Engineering Research, Volume 5, Issue 11, November-2014ISSN 2229-5518π3 α{68.88(π2 ) 0.030}123(22)Substituting for π values from (15), (16) and (17) we obtain expressions for h B1 and h B2 as:hB1 α{6.37kv 0.029RePrk v /D}DPrkvhB2 α{68.88D hB hB1 hB2 hB 6.37kvD hB α kvD 0.030RePrk v /D}0.029RePrkvD(23) 68.88PrkvD(24)Applying conservation of energy to a control surfaceabout the material, we have(25)q′′convective q′′added q′′evaporationq′′convective q′′evaporationthermo-physical properties of air from Table 3.WhereTo calculate the heat transfer coefficients at various airDα porization.0.40.023Re0.8D Pr ):IJSER1) For V 1 0.66m/s25.5297 α1 691.0651 ,2) For V 3 0.92m/s34.4787 α3 970.6760 ,3) For V 5 1.20m/s44.6411 α5 1312.1077 ,Hence the average values of α is;α1 α3 α53q′′evaporation n′′A hfg q′′convectiveα1 0.03694Thereforeα3 0.03552Whereα5 0.03402 hA(Thf TS )λ𝑑𝑤dt𝑑𝑤dt(29)(30)λ(31)λ represents the constant drying rateand the heating fluid is at 𝑇5 , Thf 𝑇5𝑇5 𝑇𝑠 𝑇5Therefore (ℎ𝐸 A T)5 λcalculating the coefficient of heat transfer as:𝐷dtFor surface temperature, 𝑇𝑠 280CSubstituting 𝛂 into (24), we have the model equation for𝐾𝑣dwhA(Thf TS ) hA( T) 0.03550 hB 0.03550(28)may be approximated as the ���𝑜𝑛of the mass (moisture loss, dw) and the latent heat of va-flow velocities using Dittus –Boelter equation [14]KV(27)Since no heat is added, and for constant dryingThe values of α can be obtained as follows using the(hD Evaporative heatGasFig 1: Heat and Mass Transfer Analogyk 0.030RePr Dv {6.37 Pr(68.88 0.059Re)}Convective heatFlow{6.37 Pr(68.88 0.059𝑅𝑅)}2.20For V 1 0.66m/s, h B1 24.5295Wm-2K-1For V 3 0.92m/s, hB3 34.4543Wm-2K-1For V 5 1.20m/s, hB5 45.5735Wm-2K-1Hence the modeled mean heat transfer coefficient is:hB (hB1 hB3 hB5 )/3(26)Therefore,hB 34.8524Wm-2K-12.2.1 Experimental Materials and MethodUnripe plantains chips were used in the experiment. Themoisture content of the plantain before drying was 26.76gdry base. The moisture content was determined by periodically weighing of the sample at 3 minutes interval, forthree hours to generate 61 data points.In evaporative heating based on heat and mass transferanalogy, as the gas flow over the moist material evaporation occurs from the surface, and the energy associatedwith the phase change is the latent heat of vaporization ofthe liquid.ℎ𝐸(5) (λ 𝑑𝑤)A( T)dt 5𝑑𝑤dt(32)(33)(5)(34)When heating fluid is at 𝑇3 , Thf 𝑇3ℎ𝐸(3) (λ 𝑑𝑤)A( T)dt 3(35)When heating fluid is at 𝑇1 , Thf 𝑇1ℎ𝐸(1) (λ 𝑑𝑤)A( T)dt 1Total Area of Plantain Chips A ((36)𝜋𝐷24)(37)D diameter of Plantain Chips 0.033mFor the 6 pieces of plantain 0.198mHence A π 0.19824 0.030795m2From (34), (35) and (36), the mean experimental heattransfer coefficienthE λdwdwdw ( T)dt ( T)dt ( T)dt A1353(38)3 RESULTS AND DISCUSSIONThe results obtained from this work are presented in Table 2-4 and Figure 2-3.IJSER 2014http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 5, Issue 11, November-2014ISSN 2229-5518Where V and T are the average values. Substituting the(hD thermo-physical properties and VBcal based on the .8D Pr ) we obtain values of heat transfercoefficient shown in Table 3.equationTABLE 2CALIBRATION OF THE BATCH DRYERS/NO13.60Velocity al0.66T166.00TemperatureT266.00(OC)T66,00TABLE 3THERMO-PHYSICAL PROPERTIES AND CALCULATED VALUES OF HEAT TRANSFER COEFFICIENT USING DITTUSBOELTER ty(kg/m3)1.03831.05161.06461.07491.0982Specific Heat Capacity(m2s-2K-1 )1008.51008.31008.21008.01007.8Dynamic Viscosity( Reynolds Number (-)10070.6412325.7514661.8617348.5320216.86Prandtl Number (-)0.7020.70220.70280.7030.705Heat Transfer .64113.1. MODELLED RESULTThe values of the variables of Table 3 are used to obtain the values of 𝛑 1, 𝛑 2, 𝛑 3, as shown in Table 4.TABLE 4𝛑- 3361IJSER 2014http://www.ijser.org