Short notes for Heat transfer

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2 Furier s Law f Heat Cnductin Shrt ntes fr Heat transfer Q = Heat transfer in given directin. A = Crss-sectinal area perpendicular t heat flw directin. dt = Temperature difference between tw ends f a blck f thickness dx dx = Thickness f slid bdy = Temperature gradient in directin f heat flw. General Heat Cnductin Equatin Carterisan Crdinates (side parallel t x, y and z-directins) q g = Internal heat generatin per unit vlume per unit time t = Temperature at left face f differential cntrl vlume k x, k y, k z = Thermal cnductivities f the material in x, y and z-directins respectively c = Specific heat f the material ρ = Density f the material α = Thermal diffusivity dτ = Instantaneus time. Fr hmgeneus and istrpic material Fr steady state cnditin (Pissn s equatin) Fr steady state and absence f internal heat generatin (Laplace equatin) Fr unsteady heat flw with n internal heat generatin

3 Cylindrical Crdinates Fr hmgeneus and istrpic material, Fr steady state unidirectinal heat flw in radial directin with n internal heat generatin, Spherical Crdinates Fr hmgeneus and istrpic material Fr steady state uni-directin heat flw in radial directin with n internal heat generatin, Thermal resistance f hllw cylinders Thermal Resistance f a Hllw Sphere Heat Transfer thrugh a Cmpsite Cylinder

4 Heat Transfer thrugh a Cmpsite Sphere Critical Thickness f Insulatin: In case f cylinder, where, k 0 = Thermal cnductivity, and h = Heat transfer cefficient The drp in temperature acrss the wall and the air film will be prprtinal t their resistances, = hl/k. Steady Flw f Heat alng a Rd Circular fin ρ=πd

5 Generalized Equatin fr Fin Rectangular fin Heat balance equatin if Ac cnstant and As P(x) linear General equatin f 2 nd rder θ = c1e mx + c2e -mx Heat Dissipatin frm an Infinitely Lng Fin (l ) Heat transfer by cnductin at base Heat Dissipatin frm a Fin Insulated at the End Tip Heat Dissipatin frm a Fin lsing Heat at the End Tip

6 Fin Efficiency Fin efficiency is given by If l (infinite length f fin), If finite length f fin, Fin Effectiveness Lumped Parameter System dt dt Q = - VCp ha T Ta dt ( T Ta) ha VCp dt

7 ln ln ha VCp T Ta t C1 ha VCp T Ta t ln Ti Ta T Ta Ti Ta exp ha VCp t Nusselt Number (Nu) It is a dimensinless quantity defined as= hl/ k, h = cnvective heat transfer cefficient, L is the characteristic length k is the thermal cnductivity f the fluid. The Nusselt number culd be interpreted physically as the rati f the temperature gradient in the fluid immediately in cntact with the surface t a reference temperature gradient (T s T ) /L. Newtn s Law f Cling says that the rate f heat transfer per unit area by cnvectin is given by Temperature distributin in a bundary layer: Nusselt mdulus The heat transfer by cnvectin invlves cnductin and mixing mtin f fluid particles. At the slid fluid interface (y = 0), the heat flws by cnductin nly, and is given by

8 In dimensinless frm, Reynld Number (Re): Critical Reynld Number: It represents the number where the bundary layer changes frm laminar t turbine flw. Fr flat plate, Re < (laminar) Re > (turbulent) Fr circular pipes, Re < 2300 (laminar flw) Stantn Number (St) 2300 < Re < 4000 (transitin t turbulent flw) Re > 4000 (turbulent flw) Grashf Number (Gr) If a bdy with a cnstant wall temperature T w is expsed t a qui scent ambient fluid at T, the frce per unit vlume can be written as: ρgβ(tw t ) where ρ = mass density f the fluid, β = vlume cefficient f expansin and g is the acceleratin due t gravity.

9 β = Cefficient f vlumetric expansin = 1/T The magnitude f Grashf number indicates whether the flw is laminar r turbulent. If the Grashf number is greater than 10 9, the flw is turbulent and Fr Grashf number less than 10 8, the flw is laminar. Fr 10 8 < Gr < 10 9, It is the transitin range. Prandtl Number (Pr): Fr liquid metal, Pr < 0.01 Fr air and gases, Pr 1 Fr water, Pr 10 Fr heavy il and grease, Pr > 10 5 Fr Pr << 1 (in case f liquid metals), the thickness f the thermal bundary layer will be much mre than the thickness f the mmentum bundary layer and vice versa. The prduct f Grashf and Prandtl number is called Rayleigh number. Or, Ra = Gr Pr Rayleigh Number (Ra) Free r natural cnvectin 10 4 < Ra < 10 9 (laminar flw) Ra > 10 9 (turbulent flw) Turbulent flw ver flat plate

10 Nu x = (Re) 0.8 (Pr) 0.33 Turbulent flw in tubes Nu = (Re) 0.8 (Pr) n where, n = 0.4 if fluid is being heated, = 0.3 if fluid is being cled. Empirical Crrelatin fr Free Cnvectin Heated surface up r cled surface dwn Laminar flw < Gr.Pr < Nu = 0.54 (Gr Pr) 0.25 Turbulent flw < Gr.Pr < Nu = 0.14 (Gr Pr) 0.33 Heated surface dwn r cled surface up Laminar flw < Gr.Pr < Nu = 0.27 (Gr Pr) 0.25 Turbulent flw < Gr.Pr < Nu = (Gr Pr) 0.33 Vertical plates and Large cylinder Laminar flw

11 10 4 < GrPr < 10 9 Nu = 0.59 (GrPr) 0.25 Turbulent flw 10 9 < GrPr < Nu = 0.13 (GrPr) Empirical Crrelatin fr Frced Cnvectin Laminar Flw ver Flat Plate Hydrdynamic bundary layer thickness Laminar Flw ver Inside Tube Cnstant heat flux, Nu = 4.36 Fuling Factr (Rf) Fin Efficiency and Fin Effectiveness η fin = (actual heat transferred) / (heat which wuld be transferred if the entire fin area were at the rt temperature)

12 Fr a very lng fin, effectiveness: And i.e., effectiveness increases by increasing the length f the fin but it will decrease the fin efficiency. Expressins fr Fin Efficiency fr Fins f Unifrm Crss-sectin: Very lng fins: Fr fins having insulated tips Lgarithmic Mean Temperature Difference (LMTD) LMTD Capacity Rati

13 Capacity rati c = mc, where c = Specific heat Effectiveness f Heat Exchanger: If m cc n < m hc h c min = m cc c If m cc n < m hc h c min = m hc h Number f Transfer Units (NTU): U = Overall heat transfer cefficient A = Surface area C min = Minimum capacity rate If m hc h < m cc c c min = m cc c If m hc h < m cc c c min = m hc h Effectiveness fr Parallel Flw Heat Exchanger

14 Effectiveness fr the Cunter Flw Heat Capacity: Heat Exchanger Effectiveness Relatin: Cncentric tube: Parallel flw: Cunter flw: Crss flw (single pass): Bth fluids unmixed: C max mixed, C min unmixed: C min mixed, C max unmixed: Ttal Emissive Pwer (E) It is defined as the ttal amunt f radiatin emitted by a bdy per unit time and area.

15 E = σt 4 W/m 2 σ = Stefan Bltzmann cnstant σ = W/m 2 K 4 Mnchrmatic (Spectral) Emissive Pwer (Eλ) It is defined as the rate f energy radiated per unit area f the surface per unit wavelength. Emissin frm Real Surface The emissive pwer frm a real surface is given by E = εσat 4 W ε = Emissivity f the surface, T = Surface temperature Emissivity (ε) It is defined as the rati f the emissive pwer f any bdy t the emissive pwer f a black bdy f same temperature. Fr black bdy, ε = 1 Fr white bdy, ε = 0 Fr gray bdy, 0< ε<1 Reflectivity (ρ) It is defined as the fractin f ttal incident radiatin that are reflected by material. Absrptivity It is defined as the fractin f ttal incident radiatin that are absrbed by material. Transmissivity It is defined as the fractin f ttal incident radiatin that are transmitted thrugh the material.

16 Fr black bdy α = 1, ρ = 0, τ = 0 Fr paque bdy τ = 0, α + ρ = 1 Fr white bdy ρ = 1, α = 1 and τ = 0 Kirchff s Law The emissivity ε and absrptivity α f a real surface are equal fr radiatin with identical temperature and wavelength. Emissive pwer f a black bdy is directly prprtinal t the furth pwer f its abslute temperature. Eb = σt 4 E b = Emissive pwer f a black bdy, σ = Stefan Bltzmann cnstant ( W/m 2 K 4 ), T = Abslute temperature f the emitting surface, K. Wien s Displacement Law Wien s displacement law state that the prduct f λ max and T is cnstant. λmax T = cnstant λ max = Wavelength at which the maximum value f mnchrmatic emissive pwer ccurs. Gray Surfaces The gray surface is a medium whse mnchrmatic emissivity (ε λ) des nt vary with wavelength. ελ = Eλ / Eλ, b But, we knw the fllwing.

17 Therefre, View Factrs: Define the view factr, F 1-2, as the fractin f energy emitted frm surface 1, which directly strikes surface 2. Reciprcity: Planck s Law: Planck suggested fllwing frmula, mnchrmatic emissive pwer f a black bdy. Ttal emissive pwer Electrical Netwrk Apprach fr Radiatin Heat Exchange

18 New Gray Bdy Factr E 1 = Emissivity fr bdy 1 E2 = Emissivity fr bdy 2 In case f black surfaces, ε1 = ε2 = 1, (Fg)12 = F1-2 In case f parallel planes, A1=A2 and F1-2 = 1 In case f cncentric cylinder r sphere, F1-2 = 1 Where, (fr cncentric cylinder) (fr cncentric sphere) When a small bdy lies inside a large enclsure Radiatin Shield

19 Radiatin netwrk fr 2 parallel infinite places separated by ne shield If ε 1 = ε 2 = ε 3 Then, and

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