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Heat-transfer-formulas

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Technological University of the Philippines Manila**We aren't endorsed by this school
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MECHANICAL 14
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Mechanical Engineering
Date
Dec 19, 2024
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9
Uploaded by ProfessorGoose4805
Heat-Transfer-FormulasThermodynamics (University of Perpetual Help System DALTA)Scan to open on StudocuStudocu is not sponsored or endorsed by any college or universityDownloaded by bronze Absole (bronze.absole@gmail.com)
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lOMoARcPSD|39380399Downloaded by bronze Absole (bronze.absole@gmail.com)Q =kA (t1− t2)xConduction – A mode of heat transfer in which heat is transferred by molecularinteraction through bodies in contact.Convection – A mode of heat transfer in which heat is transferred due to mixingand motion of particles of a substance.A.Free Convection – The substance moves because of the decrease in itsdensity which is caused by increase in temperatureB.Forced Convection – The substance moves because of the application ofmechanical power such as that of a fan.Radiation – Is a mode of heat transfer in which heat is transferred between bodiesby energy propagating electromagnetic waves.CONDUCTIONCONDUCTION THROUGH A PLANE WALL:For steady state, unidirectional flow of heat through a homogenous plane wall,Fourier’s equation gives the heat by conduction as:Where:QA==heat transmitted, Wheat transfer area, m2t1=surface temperature on hot side, ˚C or Kt2x==surface temperature on cold side, ˚C or Kthickness of the wall, mk=thermal conductivity𝐰𝐦 −˚𝐂𝐖or𝐦 − 𝐊CONDUCTION THROUGH COMPOSITE PLANE WALLS:For a composite wall shown in the figure, if the heat flows in series first throughone slab and then another, Fourier’s equation can be applied as:k1A (t1− t2)Q1 =Q2 =Q3 =x1k2A (t2− t3)x2k3A (t3− t4)xkn= thermal conductivityRT= overall resistance𝐰𝐦 − ˚𝐂𝐖or𝐦 − 𝐊3Q = Q1 = Q2 = Q3 → for steady state heat transferRT =x1 + x2k1k2+ x3k3Q = AΔt=A(t1− t4)RTx1 + x2 + x3k1k2k3Where:RT = overall resistanceQ = Q1 = Q2 = Q3 → for steady state heat transferQn=heat transmitted, WA=heat transfer area, m2Δt=temperature difference, ˚C or Ktn=surface temperature on hot side, ˚C or Ktn+1=surface temperature on cold side, ˚C or Kxn=thickness of the wall, m
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Qi = h1 A (ti − t1)Qo = ho A (t4 − to)lOMoARcPSD|39380399ho= surface film conductance on cold side𝐦𝟐 ˚𝐂Q = AΔtRTCONDUCTION FROM FLUID TO FLUIDQnA==heat transmitted, Wheat transfer area, m2Δt=temperature differencetn=surface temperature on hot side, ˚C or Ktn+1=surface temperature on cold side, ˚C or Kxn=thickness of the wall, mkn= thermal conductivityRT= overall resistance𝐰𝐦 − ˚𝐂𝐖or𝐦 − 𝐊𝐖hi= surface film conductance on hot side𝐦𝟐˚𝐂𝐖Q3 =k3A (t3− t4)x3Q = AΔt=A (ti− to)RT1+ x1+ x2+ x3+ 1hik1k2k3hoWhere:hi = surface film conductance on hot side𝐖𝐦𝟐 ˚𝐂𝐖Ho = surface film conductance on cold side𝐦𝟐 ˚𝐂OVERALL CONDUCTANCE or OVERALL COEFFICIENT OF HEAT TRANSFER, U:U =1 =1xn= thickness of the wall, m𝐰kn= thermal conductivity𝐦 − ˚𝐂𝐖or𝐦 − 𝐊RT1 + x1 + x2 + x3 + 1R= overall resistancehik1k2k3hoThi= surface film conductance on hot side𝐖𝐦𝟐˚𝐂𝐖ho= surface film conductance on cold sideU= overall conductance𝐦𝟐 ˚𝐂HEAT TRANSFER IN TERMS OF OVERALL RESISTANCE RT:A=heat transfer area, m2Δt=temperature difference, ˚C or KRT=overall resistanceQ=heat transmitted, WHEAT TRANSFER IN TERMS OF OVERALL CONDUCTANCE U:Q= U A ΔtA=heat transfer area, m2Δt=temperature difference, ˚C or KRT=overall resistanceU=overall conductanceQ=heat transmitted, WCONDUCTION THROUGH PIPE:In conduction through pipe assume that heat flows in the radial direction frominside to outside surface. Fourier’s equation gives the heat loss as:Downloaded by bronze Absole (bronze.absole@gmail.com)Q=k1A(t1−t2)1x1Q= k2A (t2− t3)2x2
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Downloaded by bronze Absole (bronze.absole@gmail.com)r2r2r2r+Rr32lOMoARcPSD|39380399Q = AΔt= t2−t1t2−t1d2𝐰𝐖RTln(r1)ln( )d1k= thermal conductivity𝐦 − ˚𝐂or𝐦 − 𝐊2πkL2πkLRT= overall resistancern= radiusDT= diameterL= lengthQ= total amount of heat that passes through the layers, WWhere:R = resistance to heat flowln(r2)R =r12πkLQ1 =t2− t1r2Q2 =t2− t3r3A= heat transfer area, m2Δt= temperature differencetn= surface temperature on hot side, ˚C or Ktn+1= surface temperature on cold side, ˚C or Kln(r1)2πk1Lln( 2 )2πk2Lkn= thermal conductivityRT= overall resistancern= radiusL= length𝐰 𝐦 − ˚𝐂𝐖or𝐦 − 𝐊Q = Δt =t1−t33Q= total amount of heat that passes through the layers, WQ1= heat passes through layer ①RTln(r1)ln( 2 )Q= heat passes through layer ②22πk1L2πk2LNote: For steady state heat transfer:Q = Q1 = Q2Where:Q1 = heat passes through layer ①Q2 = heat passes through layer ②Q = total amount of heat that passes through the layersCONDUCTION FROM FLUIDS THROUGH PIPES:Qi = hi Ai (t1 – ti)Ai=heat transfer on inside area, m2Ao=heat transfer on outside area, m2𝐖hi= surface film conductance on hot side𝐦𝟐 ˚𝐂𝐖ho= surface film conductance on cold side𝐦𝟐 ˚𝐂ti= surface temperature on hot side, ˚C or Kto= surface temperature on cold side, ˚C or KΔt= temperature differencetn= surface temperature on hot side, ˚C or Ktn+1= surface temperature on cold side, ˚C or KΔtt2− t3𝐰𝐖kn= thermal conductivityorQ2 ==2r3ln(r2)RT= overall resistance𝐦 − ˚𝐂𝐦 − 𝐊2πk2LQ = Δt =ti− torn= radiusL= lengthQ= total amount of heat that passes through the layers, WQi= heat transmitted on hot side, WQo= heat transmitted on cold side, WRT1Aihiln( )+r12πk1Lln(r )1+ 2πk2L Aoho=Qo = ho Ao (t3 – to)Q= Δt=t1− t2 1R1ln()r2r1A=heat transfer area, m2Δt=temperature differencetn=surface temperature on hot side, ˚C or Ktn+1=surface temperature on cold side, ˚C or K
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Downloaded by bronze Absole (bronze.absole@gmail.com)Q = UO AO ΔtlOMoARcPSD|39380399Note: For steady state heat transfer:Q = Q1 = Q2Where:A1 = 2π r1 LA2 = 2π r2 Lhi = surface conductance on inside surfaceho = surface conductance on outside surfaceHEAT TRANSFERRED IN TERMS OF THE OVERALL CONDUCTANCE:Q = Ui Ai ΔtorHEAT EXCHANGER- is any device which affects the transfer of heat from one substance to another.EXAMPLES of HEAT EXCHANGERS:① Steam Boiler⑤ Evaporators② Steam Condenser⑥ Economizer③ Water Heater⑦ Fluid Heaters and Coolers④ Oil Heaters⑧ Tube TanksCLASSIFICATION OF HEAT EXCHANGERS:1.Heat Exchangers wherein a fluid at constant temperature gives up heat to acolder fluid the temperature of which gradually increases as it flows throughthe device. The heating fluid can be at rest or moving in any direction. Anexample of this type would be a steam condenser.AiAo==heat transfer on inside area, m2heat transfer on outside area, m2Δt=temperature difference, ˚C or KUi=overall conductance based on inside areaUo=overall conductance based on outside areaQ=heat transmitted, W
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lOMoARcPSD|39380399Downloaded by bronze Absole (bronze.absole@gmail.com)2.Devices wherein a fluid at constant temperature receives heat from a warmerfluid the temperature of which decreases as it flows through the exchanger.The heated fluid can be at rest or moving in any direction. An example of thistype is a steam boiler.3.Parallel flow heat exchanger wherein the fluids flow in the same direction andboth of them change their temperature. Examples of this type are waterheater, oil heater and coolers.4.Counterflow heat exchangers wherein the fluids flow in directions opposite toone another. This possibly the most favorable kind of fluid heaters andcoolers.5.Cross-flow heat exchangers in which one fluid at an angle to the second oneas in the case in tube tanks.
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(lOMoARcPSD|39380399Downloaded by bronze Absole (bronze.absole@gmail.com)AMTD =(Δt)max − (Δt)min2LOGARITHMIC MEAN TEMPERATURE DIFFERENCE, LMTDLMTD =(Δt)max − (Δt)minln (Δt)max)(Δt)minARITHMETIC MEAN TEMPERATURE DIFFERENCE, AMTDRADIATION- is the mode of heat transfer through electromagnetic wave. Anything whosetemperature is above the surrounding will always radiate of significant amount. TheStefan-Boltzmann Law (otherwise known as 4th Power Law) of heat transfer governsradiation heat transfer.The Radiant heat exchange between two surfaces can be computed from Stefan-Boltzmann Law:Q4t = e σ A [(T1)] − [(T2)4] WattsWhere:Q = heat transmitted by radiation per unit time 𝐉 or 𝐖t𝐬e = emissivity factor (from 0 to 1.0)Wσ = 5.67 x 10-8m2 • K4A = radiating surface areaT1 = absolute temperature of surface radiating the heat, KT2 = absolute temperature of surface receiving the heat, KTHE CONCEPT OF A PERFECT BLACK BODYPerfect Black Body is a body that absorbs all electromagnetic radiation. Itabsorbs all wavelength such no reflection occurs.When radiant energy falls on a body, part may be absorbed, part reflected,and the remainder transmitted through the body. In mathematical form;a + r + t = 1where:a = absorptivity or the fraction of the total energy absorbedr = reflectivity or the fraction of the total energy reflectedt = transmitted or the fraction of the total energy transmitted through the body
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NRE= VDμk= VDμd ρlOMoARcPSD|39380399Downloaded by bronze Absole (bronze.absole@gmail.com)NPr=μd CpkNNU= hDkPLANCK’S LAW-All substances emit radiation, the quantity and quality of which depends upon theabsolute temperature and the properties of the material, composing the radiatingbody.KIRCHOFF’S LAW-For bodies in thermal equilibrium with their environment, the ratio of totalemissive power to the absorptivity is constant at any temperature.STEFAN BOLTZMANN LAW-The total energy emitted by a black body is proportional to the fourth power tothe absolute temperature of the body.CONVECTION-is the mechanism of heat transfer whereby heat energy is transferred by movingfluids.①Most important dimensionless group in the analysis of heat convection:A.REYNOLDS NUMBER, NREReynold’s number is a dimensionless number which is significant in the designof a model of any system in which the effect of viscosity is important incontrolling the velocities or the flow pattern of a fluid’ equal to the product ofdensity, of velocity and characteristics length divided by the fluid viscosity.Where:𝐦V= velocity,𝐬D= diameter used as characteristic length, m𝛍𝐤= kinematic viscosity,𝐬(where: μk = ρ )𝛍𝐝= dynamic viscosity, Pa-s𝐤𝐠ρ= density,𝐦𝟑B.PRANDLT NUMBER, NPrPrandlt number is a dimensionless number used in the study of forced andfreeconvection, equal to the dynamic viscosity times the specific heat atconstantpressure divided by the thermal conductivity.Where:𝛍𝐝= dynamic viscosity, Pa-s𝐂𝐩= specific heatk= thermal conductivityC.NUSSELT NUMBER, NNUNusselt number is a dimensionless number used in the study of forcedconvection which gives a measure of the ratio of the total heat transfer toconductive heat transfer, and is equal to the heat-transfer coefficient timesthecharacteristic length divided by the thermal conductivity.Where:h= heat transfer coefficientD= diameter used as characteristic lengthk= thermal conductivityD.GRASHOF NUMBER, NGRGrashof number is a dimensionless number used in the study of the freeconvection of a fluid caused by a hot body. It is equal to the product of thefluid’s coefficient of thermal expansion, the temperature difference betweenthe hot body and the fluid, the cube of a typical dimension of the body andthe
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NGR2 2= D ρ β g ∆tμd2lOMoARcPSD|39380399Downloaded by bronze Absole (bronze.absole@gmail.com)square of the fluid’s density divided by the square of the fluid’s dynamicviscosity.Where:D= diameter used as characteristic lengthρ= density of the fluidβ= coefficient of thermal expansion∆t= temperature difference between the surface and the fluidg= gravitational acceleration𝛍𝐝= dynamic viscosity of the fluid②Convective Heat transfer with known specific heat:Where:m= mass flow rate,Cp= specific heat,𝐤𝐠𝐬𝐉𝐤𝐠 − ˚𝐂∆t= temperature difference③Surface ConvectionWhere:hC= surface coefficient associated with convection,A= heat transfer area, m2t1= hot surface temperature, ˚Ct2= fluid temperature, ˚C𝐖𝐦𝟐ecoursesonline.iasri.res.in/mod/page/view.php?id=2340Q = m Cp ∆t = mCp (t2 – t1) WattsQ = hc A ∆t = hc A (t1 – t2) Watts
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