Size: px
Start display at page:

Download ""

Transcription

1

2

3

4

5

6

7

8

9

10

11

12 INDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: -- AN No. of Students: 5 Sub. No.: ME64/ME64 Time: Hours Full Marks: 6 Mid Autumn Semester Examination Sub. Name: Convective Heat and Mass Transfer Instructions: Attempt all questions. Symbols have their usual meanings. Please explain your work carefully. Clearly indicate the coordinate system used in your analysis. Make suitable assumptions wherever necessary. Please state your assumptions clearly. The following information may be useful: The divergence theorem implies that a B j nj d A = R a j x j dv, where a( x, x, x3, t ) is a continuous vector field in a region R bounded by a surface B. The divergence of a vector field, v = vr er + vφ eφ + vz e z, in cylindrical coordinates is given by v = vφ vz (rvr ) + +. r r r φ z The gradient, aplacian and material derivative of a scalar function, T (r, φ, z, t ), in cylindrical coordinates, are given by T = T T T er + eφ + ez, r r φ z T T T T= +, r + r r r r φ z DT T T vφ T T + vr + + vz. t r r φ z The viscous dissipation function in cylindrical coordinates is given by v vφ vr vz vz vr vz vφ vr vφ vφ r Φ = μ z r z r φ z r φ r r r r φ r (a) Starting with Cauchy s equations of motion, ρ τ Dvi = ρ gi + j i, derive an equation representing x j local balance of mechanical energy. Identify the terms in your equation that represent rate of change Dxi is the component of the fluid velocity, v, in the xi -direction, g i is the component of the gravitational acceleration, g, in the D direction of the coordinate axis xi, τ i j is the stress tensor, t is the time and denotes material of kinetic and potential energy. Here, ρ is the density of the fluid, vi = derivative. (b) Consider a material volume of fluid (closed system) that is translating, rotating and deforming in a gravitational force field. The net rate at which work is done on the system by surface forces exerted by the surrounding fluid is W s = S (n, r, t )v da = v τ i Bm i i ji n j da, where Bm is the surface bounding the Bm material region Rm, r = xe + xe + x3e 3 denotes the position vector of a point, ei is the unit vector in the xi -direction, n is the unit outward normal at a point on the material surface Bm and Si (n, r, t ) is the

13 stress vector at a point on a plane normal to n. Using the Gauss divergence theorem and your answer to part (a), show that W s may be expressed as W s = w dv, where w s s is the sum of four groups of terms: Rm w s = ρ D D ( v v) + ρ ( g r ) p.v + π ji dij, Term I Term II Term III Term IV p is the thermodynamic pressure, π ij is the deviatoric stress tensor and dij is the deformation tensor or rate of strain tensor. (c) Identify the term or terms (I to IV) in your answer to part (b) that result in a change in the internal energy of the system. Give reasons to justify your answer. (d) Identify the term or terms (I to IV) in your answer to part (b) that lead to a generation in entropy. (e) Identify the terms (I to IV) in your answer to part (b) that represent reversible modes of work and the terms that represent irreversible modes of work. Give reasons to justify your answer. [ = marks]. The energy equation may be expressed in cv form as ρ cv DT βt = k T + q + Φ v, κt (I) where T is the temperature, cv is the specific heat capacity of the fluid at constant volume, k is the thermal conductivity of the fluid, Φ is the viscous dissipation, q is the local rate of volumetric heat generation, β is the coefficient of thermal expansion of the fluid and κ T is the isothermal compressibility of the fluid, or in c p form as ρcp DT Dp = k T + q + Φ + β T, (II) where c p is the specific heat capacity of the fluid at constant pressure. Consider low Mach number flow of a calorically perfect gas. It was shown in the class that the viscous dissipation term and the term involving Dp may be neglected in comparison with the other terms in equation (II) when the Mach number is small, so that equation (II) may be approximated as ρcp DT = k T + q. (III) For low Mach number flows of a gas, the continuity equation may be approximated by v =. (IV) The relation (IV) suggests that equation (I) may be approximated, for low Mach number flows, by ρ cv DT = k T + q. (V) Since c p and cv for gases are not equal, the two approximate equations, (III) and (V), are not identical. Which of the two equations, equation (III) or equation (V), represents the correct approximation? Give reasons to justify your answer. Suggest an explanation for this energy equation paradox. [5 marks] 3. Consider plane Couette flow of a constant-property Newtonian-Fourier fluid between two large parallel horizontal plates that are separated by a distance. The top plate moves in its own plane, with speed U. The bottom plate is stationary. There is no externally applied pressure gradient. The velocity profile is given by u = U ( y / ), where u is the component of velocity in the x-direction and y is the coordinate normal to the walls, measured from the bottom plate. The streamlines are parallel to the x

14 axis. The fluid liberates thermal energy uniformly at a constant rate, S, per unit volume. The bottom plate is maintained at a constant temperature T. The top plate is maintained at a constant temperature T. (a) Starting with an appropriate form of the thermal energy equation, derive an ordinary differential equation for the temperature distribution T ( y ). Include the effect of viscous dissipation and internal heat generation in your analysis. (b) Express the equation derived in part (a) in non-dimensional form, using the following non- T T y, Y =. Identify the non-dimensional parameters governing the T T non-dimensional temperature distribution θ (Y ). dimensional variables: θ = (c) Obtain appropriate boundary conditions for the non-dimensional equation of part (b). (d) Determine the non-dimensional temperature distribution θ (Y ). (e) Determine the Nusselt number, Nu = h, at the bottom plate, where h is the heat transfer coefficient k based on the temperature difference between the two plates and k is the thermal conductivity of the fluid. [ =5 marks] 4. Consider the flow of a constant-property Newtonian-Fourier fluid through a circular duct of inside radius r, surrounded by air at a constant ambient temperature T. The axial velocity distribution in the r hydrodynamically developed region is given by vz = w(r ) = wav ( ), where r is the radial r coordinate and wav is the average velocity. The radial component of velocity, vr, and the swirl or azimuthal component of velocity, vφ, are zero. Assume that the temperature distribution is a function of the radial coordinate only. The effective external heat transfer coefficient, he, may be assumed to be constant. The walls of the duct may be assumed to be perfectly conducting. (a) Starting with an appropriate form of the thermal energy equation, derive an ordinary differential equation for the temperature distribution T (r ). Include the effect of viscous dissipation in your analysis. (b) Write appropriate boundary conditions for solving the ordinary differential equation of part (a). (c) Determine the radial distribution of temperature, T (r ), in the fluid. (d) Determine the wall temperature, Tw, using your answer to part (c). (e) Determine the centre-line temperature, T, using your answer to part (c). (f) Determine the limiting form of the temperature distribution when the Biot number, Bi = he (r ), is k very large, that is, Bi >>. Here, k is the thermal conductivity of the fluid flowing through the duct. (g) What physical situation does the limiting case, Bi >>, or Bi, represent? (h) What physical situation does the limiting case, Bi <<, or Bi, represent? (i) Determine the limiting form of the temperature distribution when the Biot number is very small, that is, Bi <<. Comment on your answer. Suggest a physical explanation for the behavior of the temperature field for this limiting case. (j) Determine the Nusselt number, Nu = as h = h(r ), where h is the internal heat transfer coefficient, defined k qw. Here, qw is the heat flux into the fluid, at the inside surface, r = r, of the duct. Tw T [ = marks]

15 INDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: 5-4- AN No. of Students: 47 Sub. No.: ME64/ME64 Time: 3 Hours Full Marks: End Spring Semester Examination Sub. Name: Convective Heat and Mass Transfer Instructions: Attempt all questions. Symbols have their usual meanings. Please explain your work carefully. Clearly indicate the coordinate system used in your analysis. Make suitable assumptions wherever necessary. Please state your assumptions clearly. The following information may be useful:, m n sin(mπ Y )sin(nπ Y )dy =, m = n, where m and n are integers. 6 I m = 6η ( η )η m dη =, when m >. (m + )(m + 3). Consider steady laminar fully developed forced-convection flow of a constant-property Newtonian Fourier fluid between two large parallel plates, driven by a constant applied pressure gradient in the xdirection. There is a constant heat flux, q, into the fluid from the bottom plate. The top plate is insulated. The fully developed velocity distribution is given by u = 6uavη ( η ), where u is the component of velocity in the x-direction, uav is the average velocity through the channel, η = y, is the distance between the plates and y is the coordinate normal to the plates, measured from the bottom plate. The plates have infinite span (in the z-direction). The effect of viscous dissipation and axial conduction in the fluid are negligible. (a) Show that the bulk temperature of the fluid is given by Tb = 6η ( η )T dη, where T is the temperature of the fluid. (b) Obtain an expression for the bulk temperature gradient, dtb, by integrating the energy equation across dx the channel. T Tw is a sole function of the transverse coordinate, Tb Tw q η. Using this relation, show that the local heat transfer coefficient at the bottom plate, h =, is a Tw Tb constant in the thermally developed region. Here, Tw ( x) is the temperature of the bottom plate. dtw dtb T dtb (d) Using the result of part (c), show that (i) = and (ii) = in the thermally developed dx dx x dx (c) In the thermally developed region of the flow, region. (e) Determine the temperature distribution in the fluid relative to the temperature of the bottom plate, in the thermally developed region of the flow. (f) Determine the Nusselt number at the bottom plate. [ = marks]. Consider steady laminar two-dimensional incompressible forced-convection boundary-layer flow of a constant-property Newtonian fluid over a semi-infinite flat plate aligned with the direction of a uniform isothermal oncoming free-stream. The plate is maintained at a constant temperature, Tw, higher than the temperature, T, of the free-stream. The boundary-layer equations admit a similarity solution of the form

16 v T Tw u = F (η ), V = Re/ = = G (η ), [η F (η ) F (η )], θ = u u T Tw x where F (η ) is the solution of the Blasius equation for flat plate flow, F + FF =, with boundary conditions F () = F () =, F ( ) =, and G (η ) is the solution of the equation, G + Pr F (η )G =, u Y with boundary conditions G () = and G ( ) =. Here, η =, x = x, Y = y Re/, Re =, ν x y y =, u is the component of fluid velocity in the x -direction, v is the component of fluid velocity in the y -direction, T is the fluid temperature, x is the coordinate along the plate, y is the coordinate normal to the plate, measured from the plate, u is the free-stream speed, is a reference length, ν is the kinematic viscosity of the fluid and Pr is the Prandtl number of the fluid. The values of F, F and F at u= selected values of η are given in the table below. η F (η ) F (η ) F (η ) (a) Obtain a relation between the local Nusselt number, Nu x, and the local Reynolds number, Re x. (b) The non-dimensional thickness, δ T = δt, of the thermal boundary layer varies with x as δt = Re / a x m, where a and m are constants, and δt is the dimensional thermal boundary-layer thickness. Use the similarity solution to determine the value of m. (c) Use your answer to part (a) and the above table to determine the value of Nu x Re x / when Pr =. Give a rigorous mathematical justification for your answer. (d) The value of the constant, a, in part (b) depends on the criterion used to define the thickness of the thermal boundary layer. Using the 99% criterion and the above table, estimate the value of a when Pr =. Give reasons to justify your answer. [ =5marks] 3. Consider steady laminar two-dimensional incompressible forced-convection boundary-layer flow of a large Prandtl number constant-property Newtonian fluid over a semi-infinite flat plate aligned with the direction of a uniform isothermal oncoming free-stream. The plate is maintained at a constant temperature, Tw, higher than the temperature, T, of the free-stream. (a) Estimate the order of magnitude of the non-dimensional velocity, u = u, inside the thermal boundary u layer. Express your answer in terms of the non-dimensional thickness, δ T = layer and the non-dimensional thickness, δ = δ δt, of the thermal boundary, of the hydrodynamic boundary layer. Here, u is the component of fluid velocity parallel to the plate, u is the free-stream speed, δ is the dimensional thickness of the hydrodynamic boundary layer, δ T is the dimensional thickness of the thermal boundary layer, and is a reference length.

17 (b) The ratio of the thicknesses of the thermal and hydrodynamic boundary layers, Prandtl number and is given by δt, depends on the δ δt = m in the limit as Pr, where m is a constant and Pr is the δ Pr Prandtl number of the fluid. Use scaling arguments to determine the value of m. (c) The local Nusselt number at distance from the leading edge varies as Nu ~ Re n Pr q when Pr >>, where n and q are constants, Re = u ν and ν is the kinematic viscosity of the fluid. Use scaling arguments to determine the values of n and q. [+4+4= marks] 4. Consider steady laminar two-dimensional natural-convection boundary-layer flow of a Newtonian fluid along a semi-infinite heated vertical flat plate kept at a constant temperature, Tw, higher than the ambient temperature, T. The boundary-layer equations admit a similarity solution of the form u v T T / = G (η ), η F (η ) 3F (η )], θ = = ( 4 x ) F (η ), V = Gr/ 4 = / 4 [ u u Tw T ( 4x) where x = x, u = g β (Tw T ), Gr = g β (Tw T ) 3 /ν, F (η ) and G (η ) are solutions of the equations F + 3FF ( F ) + G =, G + 3Pr F (η )G =, subject to the boundary conditions y Y F () = F () =, G () =, F ( ) =, G ( ) =, η =, Y = y Gr/ 4, y = g is the / 4 (4 x) u= gravitational acceleration, β is the coefficient of thermal expansion at the ambient temperature, ν is the kinematic viscosity of the fluid, u is the component of fluid velocity in the x -direction, v is the component of fluid velocity in the y -direction, T is the fluid temperature, x is the coordinate along the plate, pointing vertically upwards, y is the coordinate normal to the plate, measured from the plate, and is a reference length. The following table gives the values of F, G and their derivatives for Pr =, at selected values of η : η F (η ) F (η ) F (η ) G (η ) G (η ) (a) Show that Nu x = c Grx/ 4, where Nu x is the local Nusselt number, Grx is the local Grashof number and c is a constant that depends on the Prandtl number. (b) Use the relation between the local Nusselt number and the local Grashof number given in part (a) to obtain an expression for the total heat transfer rate, Q, over a length of the plate from the leading edge, per unit width in the spanwise direction (z-direction). Express your answer in terms of the constant c defined in part (a). (c) Show that the transverse velocity at the edge of the boundary layer is v = u Gr / 4b x / 4, where b is a constant that depends on the Prandtl number. Use the fact that F (η ) at a rate faster than η as η. (d) Use the above table to determine the values of b and c when Pr =. (e) Consider natural convection along a semi-infinite porous plate, with a suction velocity, V ( x,) = Kx m applied at the wall, where K and m are constants, and K is positive.

18 (i) Determine the value of m for which a similarity solution of the form indicated above exists. (ii) Modify the boundary conditions on F and G to account for the effect of suction at the porous plate when m takes the value determined in part (i). [ = marks] 5. A low Prandtl number constant-property Newtonian fluid enters the gap between two large heated parallel plates. The distance between the plates is. Both the plates are maintained at constant temperature Tw. The temperature of the fluid at the inlet is Ti. The speed of the fluid at the inlet is U. Here, Tw, Ti and U are constants, and Tw Ti. For low Prandtl number fluids, the temperature profile in a duct develops more rapidly than the velocity profile. In such a situation, the temperature profile can be determined based on a uniform velocity profile. This is called the slug flow solution. Assume that the velocity distribution in the channel can be approximated by u = U, where u is the component of velocity in the direction parallel to the plates. The effect of viscous dissipation and axial conduction in the fluid may be neglected. (a) Write the energy equation describing the temperature distribution in the fluid for the physical situation described above, in Cartesian coordinates (x, y), where x is the coordinate along the plates and y is the coordinate normal to the plates. (b) Non-dimensionalize the equation of part (a) using the following non-dimensional variables: θ= T Tw y x U, Y =,X =, where Pe = and α is the thermal diffusivity of the fluid. Pe α Ti Tw (c) Write appropriate boundary conditions for solving the partial differential equation of part (b). (d) Use the method of separation of variables to obtain the solution, θ ( X, Y ), of the partial differential equation derived in part (b), subject to the boundary conditions of part (c). (e) Use the series solution of part (d) to obtain an expression for the non-dimensional bulk temperature, θb = Tb Tw. Express your answer in terms of the coefficients of the series solution of part (d). Ti Tw (f) Obtain an expression for the variation of the local Nusselt number at the bottom plate. Express your answer in terms of the coefficients of the series solution of part (d). (g) Determine the value of the Nusselt number in the thermally developed region, using your answer to part (f). [ = 5 marks] 6. Give concise answers to the following parts, with proper justification. (a) Consider hydrodynamically developed and thermally developing forced convection in a circular tube with step change in the wall temperature. A student argues that the thermal entrance length is larger in the case of forced convection of water than in the case of forced convection of air when the velocity distribution is identical for the two cases. Do you agree with the student? (b) Consider forced convection in a channel. A student argues that the effect of viscous dissipation is more important in the case of forced convection of oil than in the case of forced convection of liquid mercury when the velocity distribution is identical for the two cases. Do you agree with the student? (c) Consider forced convection from a large heated isothermal plate aligned with the direction of a uniform isothermal oncoming free-stream of air at low Mach numbers. A student argues that the heat transfer from the plate will be doubled if the free-stream speed is doubled, for the same temperature difference between the plate and the free-stream. Do you agree with the student? (d) Consider forced convection from a large heated isothermal plate aligned with the direction of a uniform isothermal oncoming free-stream of air at low Mach numbers. A student argues that the heat transfer from the plate will be doubled if the temperature difference between the plate and the free-stream is doubled and the free-stream speed is kept the same. Do you agree with the student? (e) Consider natural convection heat transfer to air from a large heated vertical isothermal plate. A student argues that the heat transfer from the plate will be doubled if the difference between the plate temperature and the ambient temperature of the surrounding air is doubled. Do you agree with the student? [++++= marks]

19 INDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: --3 AN No. of Students: 5 Sub. No.: ME64/ME64 Time: Hours Full Marks: 6 Mid Spring Semester Examination 3 Sub. Name: Convective Heat and Mass Transfer Instructions: Attempt all questions. Symbols have their usual meanings. Please explain your work carefully. Clearly indicate the coordinate system used in your analysis. Make suitable assumptions wherever necessary. Please state your assumptions clearly. The following information may be useful: The divergence theorem states that a B j nj d A = R a j x j dv, where a( x, x, x3, t ) is a continuous vector field in a region R bounded by a surface B. s v =, T p p T ρ v β v p where β = = = and κt = ρ p T v p T v T p T v κ T cp s = T p T s cv = T v T, s p =, v T T v, (a) By integrating Cauchy s equations of motion, ρ τ Dvi = ρ gi + j i, over a material region, Rm (t ), x j bounded by a material surface, S m (t ), and using the relation, d Dφ ρφ dv = ρ dv, show that dt Rm (t ) Rm ( t ) the time rate of change of mechanical energy (sum of kinetic energy and potential energy) of a material region (closed system) occupied by a specified portion of fluid (material) that is translating, rotating and deforming in a gravitational force field is given by d ρ v v g r dv = viτ ji n j da + p v dv Φ dv. dt Rm (t ) Sm ( t ) Rm ( t ) Rm ( t ) Term Term Term 3 Dxi Here, ρ is the density of the fluid, vi = is the component of the fluid velocity, v, in the xi direction, gi is the component of the gravitational acceleration, g, in the direction of the coordinate axis D xi, t is the time, denotes material derivative, r is the position vector, n j is the jth component of the unit outward normal, n, at a point on the material surface, p is the thermodynamic pressure, Φ = π ji dij is the viscous dissipation function, and τ i j, π ij and d ij = vi v j + are respectively the elements of x j xi the stress tensor, the deviatoric stress tensor, and the deformation tensor or the rate of strain tensor. (b) By integrating the internal energy form of the equation representing local balance of thermal energy, ρ De = q p v + Φ + q, over a material region, show that the time rate of change of internal energy of a material region of fluid that is translating, rotating and deforming is given by

20 d ρ e dv = q nˆ da p v dv + Φ dv + q dv. dt R ( t ) Sm ( t ) Rm ( t ) Rm ( t ) Rm ( t ) m Term 4 Term 5 Term 6 Term 7 Here, e is the specific internal energy, q is the heat flux vector and q is the local rate of volumetric heat generation. (c) Using your answers to parts (a) and (b), obtain an equation for the time rate of change of total energy (sum of mechanical energy and internal energy) of a material region of fluid that is translating, rotating and deforming in a gravitational force field. (d) Identify the term or terms (Term to Term 7) that result in a change in the total energy of the system. Give reasons to justify your answer. (e) Identify the term or terms (Term to Term 7) that represent irreversible conversion of mechanical energy to internal energy. Give reasons to justify your answer. (f) Identify the term or terms (Term to Term 7) that represent reversible conversion of mechanical energy to internal energy or vice versa. Give reasons to justify your answer. [+++++= marks]. Consider steady forced convection of a constant-property Newtonian-Fourier fluid in the gap between two large parallel horizontal plates that are separated by a distance H. The top plate moves in its own plane, in the x-direction, with constant speed U. The bottom plate is stationary. There is no externally applied pressure gradient. The velocity field is given by u = U ( y / H ), v =, w =, where u, v and w are the components of velocity in the x, y and z directions respectively, and y is the coordinate normal to the walls, measured from the top surface of the bottom plate. The streamlines are parallel to the x-axis. The bottom plate is uniformly heated, with a constant heat flux q. Heat transfer from the top plate to the surrounding air which is at a constant ambient temperature T, may be represented by a convective boundary condition, with constant external heat transfer coefficient he. The thermal conductivity, k, of the fluid and the thermal conductivity, k w, of the walls of the channel are constant. (a) Starting with an appropriate form of the thermal energy equation, derive an ordinary differential equation for the temperature distribution, T ( y ), in the fluid. Include the effect of viscous dissipation in your analysis. (b) Starting with the equation for heat conduction in a solid, show that the transverse gradient, dts, of dy the temperature in the top plate is constant. Here, Ts ( y ) is the temperature distribution in the top plate. (c) Express the equation derived in part (a) in non-dimensional form, using the following nondimensional variables: θ = T T y, Y=. qh H k (d) Obtain an appropriate non-dimensional boundary condition for the non-dimensional equation of part (c), at the interface between the fluid and the top surface of the bottom plate ( Y = ).

21 (e) Using your answer to part (b) or otherwise, show that the boundary condition for the non-dimensional equation of part (c), at the interface between the fluid and the bottom surface of the top plate, may be expressed in non-dimensional form as dθ + Bi θ = at Y =, dy heff H where Bi is an effective Biot number defined by Bi =, that may be determined using the relation k k tw, = + Bi Bie k w H heff is an effective heat transfer coefficient, Bie is the Biot number based on the external heat transfer coefficient, given by Bie = he H, and tw is the thickness of the top plate. k (f) Without solving the equation obtained in part (c), show that the non-dimensional temperature distribution depends on the Biot number, Bi, and the modified Brinkman number, Br, defined by Br = Ec* Pr, using your answers to parts (c), (d) and (e), where Ec* is a modified Eckert number defined by Ec* = U, c p is the specific heat capacity of the fluid and Pr is the Prandtl number of the fluid. qh cp k (g) Determine an approximate form of the non-dimensional equation obtained in part (c) when the effect of viscous dissipation is negligible. Solve this approximate equation to obtain the distribution of the nondimensional temperature, θ, when the effect of viscous dissipation is negligible. (h) Without solving the equation obtained in part (c), show that the solution of the non-dimensional equation of part (c) may be expressed as θ (Y ; Bi, Br ) = θ (Y ; Bi ) + Br θ (Y ; Bi), where θ (Y ; Bi ) is the temperature distribution obtained in part (g) when the effects of viscous dissipation is negligible, using the principle of superposition. Obtain a differential equation for θ (Y ; Bi ), and appropriate boundary conditions for this differential equation. (i) Determine the function, θ (Y ; Bi ), by solving the differential equation derived in part (h), subject to appropriate boundary conditions. (j) Using your answers to parts (g), (h) and (i), obtain an expression for the temperature of the lower surface of the top plate. (k) Using your answers to parts (g), (h) and (i), determine the temperature distribution in the fluid for the special case when the inside surface of the upper plate, which is adjacent to the fluid, is at temperature T. Do not solve the equation with the boundary condition T = T at y = H, to answer this question. (l) Determine the limiting form of the temperature distribution when the Biot number is very small, that is, Bi <<. Suggest a physical explanation for the behavior of the temperature field for this limiting case. [ =3 marks]

22 3 (a) The local balance of thermal energy may be expressed in entropy form as ρt Ds = q + q + Φ, where ρ is the density of the fluid, T is the temperature, s is the specific entropy, t is the time, D denotes material derivative, q is the heat flux vector, q is the local rate of volumetric heat generation and Φ is the viscous dissipation. Starting with the above equation, use appropriate thermodynamic relations to show that the energy equation for a Fourier fluid may be expressed in cv form as ρ cv DT βt = ( k T ) + q + Φ v, κt where cv is the specific heat capacity of the fluid at constant volume, k is the thermal conductivity of the fluid, β is the coefficient of thermal expansion of the fluid, κ T is the isothermal compressibility of the fluid and v is the fluid velocity. (b) Two students, A and B, were asked to obtain an approximate form of the energy equation for low Mach number flow of a gas. Both the students, A and B, agree that the viscous dissipation term may be neglected since the Eckert number is small for low Mach number flows. Student A argues that the last term on the right hand side of the cv form of the energy equation stated in part (a), involving v, may be neglected since the continuity equation is approximated by v = for low Mach number flows of a gas, and hence, the equation may be approximated as DT = ( k T ) + q. Student B thinks that this term, involving v, cannot be neglected even if the Mach number is small. ρ cv Which student, A or B, do you think is correct? Give reasons to justify your answer. [5+5= marks].

23 INDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: AN No. of Students: 5 Sub. No.: ME64 Time: 3 Hours Full Marks: End Spring Semester Examination 3 Sub. Name: Convective Heat and Mass Transfer Instructions: Attempt all questions. This question paper consists of four pages. Symbols have their usual meanings. Please explain your work carefully. Make suitable assumptions wherever necessary. Please state your assumptions clearly. The following information may be useful: e t dt = π, e t3 dt =.893, erfc( z ) d z = z erfc( z ) π e z + constant. limη erfc(η ) = η. Consider steady laminar forced-convection flow of a low Prandtl number constant-property Newtonian Fourier fluid between two large parallel plates, driven by an applied pressure gradient in the x-direction. The distance between the plates is. There is a constant heat flux, q, into the fluid from the bottom plate. The top plate is insulated. The velocity profile at the entrance of the parallel-plate channel is uniform. The speed of the fluid at the inlet of the channel is U. For low Prandtl number fluids, the temperature profile in a duct develops more rapidly than the velocity profile. In such a situation, the temperature profile can be obtained based on a uniform velocity profile. This is called the slug flow solution. Assume that the velocity distribution in the channel can be approximated by u = U, where u is the component of velocity in the x-direction. The plates have infinite span (in the z-direction). The effect of viscous dissipation and axial conduction in the fluid may be neglected. (a) Show that the bulk temperature of the fluid is given by Tb = T dη, where T is the temperature of the y fluid, η =, y is the coordinate normal to the plates, measured from the bottom plate. dt (b) Obtain an expression for the bulk temperature gradient, b, by integrating the energy equation across dx the channel. T Tw is a sole function of the transverse coordinate, Tb Tw q η. Using this relation, show that the local heat transfer coefficient at the bottom plate, h =, is a Tw Tb constant in the thermally developed region. Here, Tw ( x) is the temperature of the bottom plate. dtw dtb T dtb (d) Using the result of part (c), show that (i) = and (ii) = in the thermally developed dx dx x dx (c) In the thermally developed region of the flow, region. (e) Determine the temperature distribution in the fluid relative to the temperature of the bottom plate, in the thermally developed region of the flow. (f) Determine the Nusselt number at the bottom plate. [ = marks]. Consider large Peclet number steady laminar axisymmetric hydrodynamically developed incompressible flow of a constant-property Newtonian Fourier fluid through a circular duct of radius r,

24 Ti, z <, where z is the axial coordinate, Tw, z with step change in wall temperature, given by T ( r, z ) = Ti and Tw are constants. The inlet fluid temperature is Ti. The non-dimensional temperature, Tw T = θ cn f n (r * )e λn Z, θ=, downstream of the discontinuity in wall temperature is given by Tw Ti n = d * df n where r * = r / r, f n (r * ) is the solution of the equation, (r ) + λn r *[ (r * ) ] f n =, with dr * dr * boundary condition f n () =, satisfying the regularity condition f n () =, Z = z /(r Pe), Pe = wav (r ) / α, wav is the average velocity and α is the thermal diffusivity of the fluid. The important constants are given in the following table. n λn cn f n () (a) Use order of magnitude analysis to obtain an estimate of the order of magnitude of the thermal entrance length for large Peclet number flows. (b) Show that * * * * r ( r ) f n (r )dr = (c) Use the θ b = 4 n = result of cn f n () e λn Z. λ n part (a) to f n () λn. show that the non-dimensional bulk-temperature is (d) Obtain an expression for the local Nusselt number. Express your answer in terms of the constants cn, λn, Z and other appropriate quantities. (e) Determine the numerical value of the Nusselt number in the thermally developed region. [ = marks] 3. Consider steady laminar two-dimensional natural-convection boundary-layer flow of a Newtonian fluid with Prandtl number of order one, along a large vertical flat plate kept at a constant temperature, Tw, higher than the ambient temperature, T. (a) Using the Boussinesq approximation, write the continuity, Navier-Stokes and energy equations describing the flow in non-dimensional form, using the following non-dimensional variables: y u v p + ρ g x T T x = x, y =,u =,v =, P =,θ=. ρ u u u ΔT

25 Here the bars denote dimensional quantities, is a reference length, u is a reference velocity, ΔT = Tw T, x is the vertical coordinate, measured from the leading edge of the plate, y is the horizontal coordinate normal to the plate, measured from the plate, u and v are the components of velocity in the x and y directions respectively, T is the temperature, p is the pressure, ρ is the density of the fluid at temperature T, and g is the gravitational acceleration. (b) Determine the reference velocity, u, by equating the order of magnitudes of the inertia and buoyancy force terms in the vertical momentum equation. (c) Use scale analysis to estimate (i) the order of magnitude, δ, of the non-dimensional thickness of the boundary layer, and (ii) the order of magnitude, v, of the non-dimensional horizontal velocity, v, inside the boundary layer. Express your answer in terms of the Grashof number Gr = g β ΔT 3 / υ, where β is the coefficient of thermal expansion at the ambient temperature and υ is the kinematic viscosity of the fluid. (d) Express the system of equations of part (a) using scaled variables, Y = y / δ, V = v / v. Obtain the boundary-layer equations that describe the flow for large values of Gr. (e) Write appropriate boundary conditions for the system of equations of part (d). (f) The boundary-layer equations for natural convection along a semi-infinite flat plate admit a similarity solution of the form ψ = x a F (η ), θ = x b G (η ), satisfying the boundary conditions of part (e), where ψ is the streamfunction defined by the equations u = ψ ψ Y, V =, and η = m. Determine the Y x x values of a, b, and m. (g) Use your answer to part (f) to show that Nu x = c Grxn, where Nu x is the local Nusselt number, Grx is the local Grashof number and c is a constant that depends on the Prandtl number. Determine the value of n. [ =5 marks] 4. Consider steady laminar two-dimensional incompressible forced-convection boundary-layer flow of a low Prandtl number constant-property Newtonian Fourier fluid over a semi-infinite flat plate aligned with the direction of a uniform isothermal oncoming flow with flow speed u and temperature T. The plate is subjected to a constant surface heat flux, q. The velocity field may be approximated by u = u, v = in the entire thermal boundary layer. Here, u and v are the components of the velocity in the x and y directions, x is a coordinate along the plate, measured from the leading edge, y is the coordinate normal to the plate, measured from the plate. (a) Write the boundary-layer energy equation using the approximate velocity field given above, in dimensional form. (b) By differentiating the boundary-layer energy equation show that the heat flux in the direction normal to the plate, q y = k T, satisfies the same differential equation as the temperature distribution. y (c) Obtain suitable boundary conditions for the partial differential equation of part (b). (d) Show that the solution of the partial differential equation of part (b), subject to the boundary conditions obtained in part (c), is given by q y = qerfc (η ), 3

26 y where η = αx, α is the thermal diffusivity of the fluid, erfc (η ) is the complementary error u function defined by erf (η ) = η e π t erfc (η ) = erf (η ), and erf (η ) is the error function defined by dt. (e) Use your answer to part (d) to determine the temperature distribution T ( x, y ). (f) Determine the variation of the wall temperature, Tw ( x), with distance along the plate. (g) Obtain an expression for the local Nusselt number. Express your answer in terms of the local Reynolds number and the Prandtl number. [ =5 marks] 5. Give concise answers to the following parts, with proper justification. (a) Convective heating or cooling may be described under certain circumstances by Newton s law of cooling, which states that the rate of heat loss of a body is directly proportional to the difference in temperatures between the body and its surroundings. Does Newton s law of cooling hold for natural convection from a large vertical isothermal plate? (b) Consider forced convection from a large heated isothermal plate aligned with the direction of a uniform isothermal oncoming free-stream of air at low Mach numbers. An engineer wants to double the rate of heat transfer from the plate by increasing the free-stream speed. By what factor should he increase the free-stream speed in order to increase the rate of heat transfer from the plate by a factor of two? (c) Consider steady laminar axisymmetric hydrodynamically developed and thermally developing forced convection of water in a circular tube with step change in the wall temperature. The Reynolds number based on the average velocity through the duct and the diameter of the tube is 5. A student argues that the thermal entrance length will be halved if the volume flow rate is doubled. Do you agree with the student? (d) Consider steady laminar axisymmetric hydrodynamically and thermally developed forced convection of water in a circular tube. The inside surface of the tube is at a constant temperature higher than the temperature of the fluid at the inlet. An engineer thinks that the heat flux at the surface of the tube in the thermally developed region of the flow will be doubled if the flow rate is doubled. Do you agree with the engineer? (e) Consider forced convection in a channel. A student argues that the effect of viscous dissipation is more important in the case of forced convection of water than in the case of forced convection of oil when the velocity distribution is identical for the two cases. Do you agree with the student? [++++= marks] 4

27 INDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: --4 AN No. of Students: 7 Sub. No.: ME64/ME64 Time: Hours Full Marks: 6 Mid Spring Semester Examination 4 Sub. Name: Convective Heat and Mass Transfer Instructions: Attempt all questions. Symbols have their usual meanings. Please explain your work carefully. Clearly indicate the coordinate system used in your analysis. Make suitable assumptions wherever necessary. Please state your assumptions clearly. The following information may be useful: The divergence theorem states that a B j nj d A = R a j x j dv, where a( x, x, x3, t ) is a continuous vector field in a region R bounded by a surface B. D Dφ dv ρ φ dv = ρ Rm ( t ) Rm ( t ). (a) Show that the rate at which work is done by surface forces acting on a material region, Rm (t), of fluid (closed system) that is translating, rotating and deforming in a gravitational force field may be expressed as W&s = Rm ( t ) w& s dv, where w& s is the sum of four groups of terms: w& s = ρ D D ( v v) + ρ ( g r ) p.v + Φ, Term I Term II Term III Term IV Φ = π ji dij, π ij is the deviatoric stress tensor, dij is the deformation tensor or rate of strain tensor, ρ is the density of the fluid, v is the fluid velocity, g is the gravitational acceleration, p is the thermodynamic D denotes material derivative and r denotes position vector. De (b) Using your answer to part (a), show that ρ = q p v + Φ, where e is the specific internal pressure, t is the time, energy and q is the heat flux vector. (c) Using your answer to part (b), show that ρt Ds = q + Φ, where T is the absolute temperature and s is the specific entropy. (d) Using your answer to part (c), show that the rate of production or generation of entropy may be expressed as P&s = p& s dv, where Rm ( t ) p& s = Φ T Term A (e) Obtain sufficient conditions on Term A ( Φ T q T. T. Term B )and Term B ( q T ) in your answer to part (d) such T that the Clausius-Duhem inequality, p& s, holds for all possible circumstances. (f) Identify the term or terms (I to IV) in your answer to part (a) that result in generation of entropy. Give reasons to justify your answer. (g) Identify the terms (I to IV) in your answer to part (a) that represent reversible modes of work and the terms that represent irreversible modes of work. Give reasons to justify your answer. [ =3 marks]

28 . Consider steady laminar forced convection of a constant-property Newtonian-Fourier fluid in the gap between two large parallel horizontal plates that are separated by a distance, with a constant applied pressure gradient in the x-direction. The bottom plate is maintained at a constant temperature T. The top plate is uniformly heated, with a constant heat flux, q, into the fluid. The fully-developed velocity field is given by u = 6uav ( y / )( y / ), v =, w =, where u, v and w are the components of velocity in the x, y and z directions respectively, uav is the average velocity through the channel and y is the coordinate normal to the walls, measured from the bottom plate. Assume that the plates have infinite span in the z-direction. Considering the effect of viscous dissipation, determine (a) the temperature distribution in the fluid, (b) the temperature of the top plate, and (c) the Nusselt number, Nu = h, at the bottom plate, where h is the heat transfer coefficient based on k the temperature difference between the two plates and k is the thermal conductivity of the fluid. [++4=5 marks] 3. Consider steady laminar forced-convective heat transfer to a low Prandtl number constant-property Newtonian-Fourier fluid in a circular duct of radius, R, subjected to uniform surface heat flux qw. The velocity profile at the entrance of the duct is uniform. The speed of the fluid at the inlet of the duct is W. For low Prandtl number fluids (e.g. liquid metals), the temperature profile in a duct develops more rapidly than the velocity profile. In such a situation, the temperature distribution can be obtained based on a uniform velocity profile. This is called the slug flow solution. Accordingly, consider thermally developed flow with a velocity distribution approximated by w = W, where w is the axial component of the fluid velocity. Neglecting the effect of viscous dissipation and axial conduction in the fluid, determine (a) the variation of the bulk temperature, Tb, of the fluid in the axial direction, (b) the variation of the temperature, Tw, of the duct wall in the axial direction in the thermally developed region of the flow, (c) the temperature distribution in the fluid, relative to the temperature, Tw, of the duct wall, in the thermally developed region of the flow, and (d) the Nusselt number based on the duct diameter. [ =5 marks]

29 INDIAN INSTITUTE OF TECHNOOGY, KHARAGPUR Date: AN No. of Students: 7 Sub. No.: ME64/ME64 Time: 3 Hours Full Marks: End Spring Semester Examination 4 Sub. Name: Convective Heat and Mass Transfer Instructions: Attempt all questions. This question paper consists of four pages. Symbols have their usual meanings. Please explain your work carefully. Make suitable assumptions wherever necessary. Please state your assumptions clearly. The following information may be useful: e ξ 3 dξ = Consider steady laminar flow of a low Prandtl number constant-property Newtonian Fourier fluid between two large parallel plates, driven by an applied pressure gradient in the x-direction. The distance between the plates is. The bottom plate is insulated. There is a constant heat flux, q, into the fluid from the top plate. The fluid flowing through the channel is heated when q > and cooled when q <. The velocity profile at the entrance of the parallel-plate channel is uniform. The speed of the fluid at the inlet of the channel is U. For low Prandtl number fluids, the temperature profile in a duct develops more rapidly than the velocity profile. In such a situation, the temperature profile can be obtained based on a uniform velocity profile. This is called the slug flow solution. Assume that the velocity distribution in the channel can be approximated by u = U, where u is the component of velocity in the x-direction. The plates have infinite span (in the z-direction). The effect of viscous dissipation and axial conduction in the fluid may be neglected. (a) Write appropriate thermal boundary conditions at the top and bottom plates. (b) Show that the bulk temperature of the fluid is given by Tb = T dη, where T is the temperature of the fluid, η = y, y is the coordinate normal to the plates, measured from the bottom plate. (c) By integrating the energy equation across the channel, or otherwise, obtain an expression for the bulk temperature gradient, dtb. dx T Tw is a sole function of the transverse coordinate, Tb Tw q η. Using this relation, show that the local heat transfer coefficient at the top plate, h =, is a Tw Tb constant in the thermally developed region. Here, Tw ( x) is the temperature of the top plate. dtw dtb T dtb (e) Using the result of part (d), show that (i) and (ii) in the thermally developed = = dx dx x dx (d) In the thermally developed region of the flow, region. (f) Determine the temperature distribution in the fluid relative to the temperature of the top plate, in the thermally developed region of the flow. (g) Determine the Nusselt number at the top plate. [ = marks]. Consider large Peclet number steady laminar axisymmetric hydrodynamically developed incompressible flow of a constant-property Newtonian Fourier fluid through a circular duct of radius R, Ti, z <, where z is the axial coordinate, Tw, z with step change in wall temperature, given by T ( R, z ) =

30 Ti and Tw are constants. The inlet fluid temperature is Ti. The axial velocity through the duct is given by w = wav ( r* ), where r * = r / R, r is the radial coordinate and wav is the average velocity through the duct. The non-dimensional temperature, θ = wall temperature is given by θ = c n = Tw T, downstream of the discontinuity in Tw Ti f (r * )e λn Z, where, f n (r * ) is the solution of the equation, n n d * df n (r ) + λn r *[ (r * ) ] f n =, with boundary condition f n () =, satisfying the regularity dr * dr * condition f n () =, Z = z /( RPe), Pe = wav ( R ) / α, and α is the thermal diffusivity of the fluid. The important constants are given in the following table. n λn cn f n () (a) Show that the non-dimensional bulk temperature is given by θb = 4 r* ( r* ) θ d r*. f n () * * * * = r r f ( r ) dr. ( ) n λn (b) Show that (c) Using the result of parts (a) and (b), show that the non-dimensional bulk-temperature may be expressed as θ b = 4 n = cn f n () λ n e λn Z. (d) Obtain an expression for the local Nusselt number. Express your answer in terms of the constants cn, λn, Z and other appropriate quantities. (e) Determine the numerical value of the Nusselt number in the thermally developed region. [ = marks] 3. Consider steady laminar two-dimensional incompressible forced-convection boundary-layer flow of a constant-property Newtonian Fourier fluid over a semi-infinite flat plate aligned with the direction of a uniform oncoming isothermal free-stream. The plate is kept at a constant temperature, Tw, higher than the ambient temperature T. The boundary-layer energy equation, u* u* θ θ θ +V = + Ec, admits * Y Pr Y x Y a similarity solution of the form θ = G (η ; Pr, Ec), where θ = Y = Re/ y*, y* = y /, u * = Ec = u, c p (Tw T ) Tw T Y, η=, x* = x /, * Tw T x u v = F (η ), V = Re/ = [η F (η ) F (η )], Re = u / υ, u u x* F (η ) is the solution of the equation F + F F =, with boundary conditions F () = F () =, F ( ) =, T is the fluid temperature, u and v are the components of the fluid velocity in the x and y directions, x is the coordinate along the plate, y is the coordinate normal to the

31 plate, u is the free-stream speed, is a reference length, υ, c p and Pr are respectively the kinematic viscosity, the specific heat capacity at constant pressure and the Prandtl number of the fluid. The origin of the coordinate system is at the leading edge of the plate. The values of F, F and F at selected values of η are given in the table below. η F (η ) F (η ) F (η ) (a) Starting with the boundary-layer energy equation stated above, derive an ordinary differential equation for G. (b) Obtain appropriate boundary conditions for the equation derived in part (a). (c) Show that the local Nusselt number predicted by the similarity solution is of the form, Nu x = c(pr, Ec) Re mx, where Re x = u x / υ is the local Reynolds number, m is a universal constant and c(pr, Ec) is a constant that depends on the Prandtl and Eckert numbers. Determine the value of m. (d) Obtain a suitable approximate form of the equation derived in part (a), which can be solved analytically, for fluids with Pr. Do not attempt to solve this equation. (e) For the special case when Pr and Ec, solve the approximate equation of part (d) without the viscous dissipation term, and show that c(pr, Ec) = b Pr n when Pr and Ec, where b and n are universal constants. Determine the values of b and n. [ =5 marks] 4. Consider steady laminar two-dimensional natural-convection boundary-layer flow of a Newtonian Fourier fluid along a semi-infinite heated vertical flat plate kept at a constant temperature, Tw, higher than the ambient temperature, T. The boundary-layer equations admit a similarity solution of the form u* = / v u T T = G (η ), = ( 4 x* ) F (η ), V = Gr/ 4 = η F (η ) 3F (η )], θ = [ / 4 u ( 4 x * ) Tw T u where x* = x, u = g β (Tw T ), Gr = g β (Tw T ) 3 / υ, F (η ) and G (η ) are solutions of the equations F + 3FF ( F ) + G =, G + 3Pr F (η )G =, subject to the boundary conditions Y y F () = F () =, G () =, F ( ) =, G ( ) =, η =, Y = Gr/ 4 y*, y* =, g is the * / 4 (4 x ) gravitational acceleration, β is the coefficient of thermal expansion at the ambient temperature, υ is the kinematic viscosity of the fluid, u and v are the components of the fluid velocity in the x and y directions, T is the fluid temperature, x is the coordinate along the plate, pointing vertically upwards, y is the coordinate normal to the plate and is a reference length. The origin of the coordinate system is at the leading edge of the plate. The following table gives the values of F, G and their derivatives for Pr =, at selected values of η : η F (η ) F (η ) F (η ) G (η ) G (η )

6.2 Governing Equations for Natural Convection

6.2 Governing Equations for Natural Convection 6. Governing Equations for Natural Convection 6..1 Generalized Governing Equations The governing equations for natural convection are special cases of the generalized governing equations that were discussed

More information

Problem 4.3. Problem 4.4

Problem 4.3. Problem 4.4 Problem 4.3 Problem 4.4 Problem 4.5 Problem 4.6 Problem 4.7 This is forced convection flow over a streamlined body. Viscous (velocity) boundary layer approximations can be made if the Reynolds number Re

More information

CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW

CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW 4.1 Introduction Boundary layer concept (Prandtl 1904): Eliminate selected terms in the governing equations Two key questions (1) What are the

More information

Chapter 7: Natural Convection

Chapter 7: Natural Convection 7-1 Introduction 7- The Grashof Number 7-3 Natural Convection over Surfaces 7-4 Natural Convection Inside Enclosures 7-5 Similarity Solution 7-6 Integral Method 7-7 Combined Natural and Forced Convection

More information

Principles of Convection

Principles of Convection Principles of Convection Point Conduction & convection are similar both require the presence of a material medium. But convection requires the presence of fluid motion. Heat transfer through the: Solid

More information

CONVECTIVE HEAT TRANSFER

CONVECTIVE HEAT TRANSFER CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 4 HEAT TRANSFER IN CHANNEL FLOW BASIC CONCEPTS BASIC CONCEPTS Laminar

More information

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

More information

Chapter 6 Laminar External Flow

Chapter 6 Laminar External Flow Chapter 6 aminar Eternal Flow Contents 1 Thermal Boundary ayer 1 2 Scale analysis 2 2.1 Case 1: δ t > δ (Thermal B.. is larger than the velocity B..) 3 2.2 Case 2: δ t < δ (Thermal B.. is smaller than

More information

Convection. forced convection when the flow is caused by external means, such as by a fan, a pump, or atmospheric winds.

Convection. forced convection when the flow is caused by external means, such as by a fan, a pump, or atmospheric winds. Convection The convection heat transfer mode is comprised of two mechanisms. In addition to energy transfer due to random molecular motion (diffusion), energy is also transferred by the bulk, or macroscopic,

More information

CONVECTIVE HEAT TRANSFER

CONVECTIVE HEAT TRANSFER CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 3 LAMINAR BOUNDARY LAYER FLOW LAMINAR BOUNDARY LAYER FLOW Boundary

More information

UNIT II CONVECTION HEAT TRANSFER

UNIT II CONVECTION HEAT TRANSFER UNIT II CONVECTION HEAT TRANSFER Convection is the mode of heat transfer between a surface and a fluid moving over it. The energy transfer in convection is predominately due to the bulk motion of the fluid

More information

ENGR Heat Transfer II

ENGR Heat Transfer II ENGR 7901 - Heat Transfer II Convective Heat Transfer 1 Introduction In this portion of the course we will examine convection heat transfer principles. We are now interested in how to predict the value

More information

Introduction to Heat and Mass Transfer. Week 14

Introduction to Heat and Mass Transfer. Week 14 Introduction to Heat and Mass Transfer Week 14 Next Topic Internal Flow» Velocity Boundary Layer Development» Thermal Boundary Layer Development» Energy Balance Velocity Boundary Layer Development Velocity

More information

Fundamentals of Fluid Dynamics: Elementary Viscous Flow

Fundamentals of Fluid Dynamics: Elementary Viscous Flow Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research

More information

The Effect Of MHD On Laminar Mixed Convection Of Newtonian Fluid Between Vertical Parallel Plates Channel

The Effect Of MHD On Laminar Mixed Convection Of Newtonian Fluid Between Vertical Parallel Plates Channel The Effect Of MH On Laminar Mixed Convection Of Newtonian Fluid Between Vertical Parallel Plates Channel Rasul alizadeh,alireza darvish behanbar epartment of Mechanic, Faculty of Engineering Science &

More information

FORMULA SHEET. General formulas:

FORMULA SHEET. General formulas: FORMULA SHEET You may use this formula sheet during the Advanced Transport Phenomena course and it should contain all formulas you need during this course. Note that the weeks are numbered from 1.1 to

More information

Fluid Dynamics Exercises and questions for the course

Fluid Dynamics Exercises and questions for the course Fluid Dynamics Exercises and questions for the course January 15, 2014 A two dimensional flow field characterised by the following velocity components in polar coordinates is called a free vortex: u r

More information

Convection Heat Transfer. Introduction

Convection Heat Transfer. Introduction Convection Heat Transfer Reading Problems 12-1 12-8 12-40, 12-49, 12-68, 12-70, 12-87, 12-98 13-1 13-6 13-39, 13-47, 13-59 14-1 14-4 14-18, 14-24, 14-45, 14-82 Introduction Newton s Law of Cooling Controlling

More information

Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay

Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay Lecture No. 18 Forced Convection-1 Welcome. We now begin our study of forced convection

More information

Parallel Plate Heat Exchanger

Parallel Plate Heat Exchanger Parallel Plate Heat Exchanger Parallel Plate Heat Exchangers are use in a number of thermal processing applications. The characteristics are that the fluids flow in the narrow gap, between two parallel

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master Degree in Mechanical Engineering Numerical Heat and Mass Transfer 15-Convective Heat Transfer Fausto Arpino f.arpino@unicas.it Introduction In conduction problems the convection entered the analysis

More information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

More information

Fundamental Concepts of Convection : Flow and Thermal Considerations. Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D.

Fundamental Concepts of Convection : Flow and Thermal Considerations. Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D. Fundamental Concepts of Convection : Flow and Thermal Considerations Chapter Six and Appendix D Sections 6.1 through 6.8 and D.1 through D.3 6.1 Boundary Layers: Physical Features Velocity Boundary Layer

More information

MYcsvtu Notes HEAT TRANSFER BY CONVECTION

MYcsvtu Notes HEAT TRANSFER BY CONVECTION www.mycsvtunotes.in HEAT TRANSFER BY CONVECTION CONDUCTION Mechanism of heat transfer through a solid or fluid in the absence any fluid motion. CONVECTION Mechanism of heat transfer through a fluid in

More information

MOMENTUM TRANSPORT Velocity Distributions in Turbulent Flow

MOMENTUM TRANSPORT Velocity Distributions in Turbulent Flow TRANSPORT PHENOMENA MOMENTUM TRANSPORT Velocity Distributions in Turbulent Flow Introduction to Turbulent Flow 1. Comparisons of laminar and turbulent flows 2. Time-smoothed equations of change for incompressible

More information

6. Laminar and turbulent boundary layers

6. Laminar and turbulent boundary layers 6. Laminar and turbulent boundary layers John Richard Thome 8 avril 2008 John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 8 avril 2008 1 / 34 6.1 Some introductory ideas Figure 6.1 A boundary

More information

Detailed Outline, M E 521: Foundations of Fluid Mechanics I

Detailed Outline, M E 521: Foundations of Fluid Mechanics I Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic

More information

Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer

Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer 1. Nusselt number Summary of Dimensionless Numbers of Fluid Mechanics and Heat Transfer Average Nusselt number: convective heat transfer Nu L = conductive heat transfer = hl where L is the characteristic

More information

Chapter 9: Differential Analysis of Fluid Flow

Chapter 9: Differential Analysis of Fluid Flow of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known

More information

HEAT TRANSFER BY CONVECTION. Dr. Şaziye Balku 1

HEAT TRANSFER BY CONVECTION. Dr. Şaziye Balku 1 HEAT TRANSFER BY CONVECTION Dr. Şaziye Balku 1 CONDUCTION Mechanism of heat transfer through a solid or fluid in the absence any fluid motion. CONVECTION Mechanism of heat transfer through a fluid in the

More information

Heat and Mass Transfer

Heat and Mass Transfer 1 Comments on six papers published by S.P. Anjali Devi and R. Kandasamy in Heat and Mass Transfer, ZAMM, Mechanics Research Communications, International Communications in Heat and Mass Transfer, Communications

More information

Mechanical Engineering. Postal Correspondence Course HEAT TRANSFER. GATE, IES & PSUs

Mechanical Engineering. Postal Correspondence Course HEAT TRANSFER. GATE, IES & PSUs Heat Transfer-ME GATE, IES, PSU 1 SAMPLE STUDY MATERIAL Mechanical Engineering ME Postal Correspondence Course HEAT TRANSFER GATE, IES & PSUs Heat Transfer-ME GATE, IES, PSU 2 C O N T E N T 1. INTRODUCTION

More information

MIXED CONVECTION OF NEWTONIAN FLUID BETWEEN VERTICAL PARALLEL PLATES CHANNEL WITH MHD EFFECT AND VARIATION IN BRINKMAN NUMBER

MIXED CONVECTION OF NEWTONIAN FLUID BETWEEN VERTICAL PARALLEL PLATES CHANNEL WITH MHD EFFECT AND VARIATION IN BRINKMAN NUMBER Bulletin of Engineering Tome VII [14] ISSN: 67 389 1. Rasul ALIZADEH,. Alireza DARVISH BAHAMBARI, 3. Komeil RAHMDEL MIXED CONVECTION OF NEWTONIAN FLUID BETWEEN VERTICAL PARALLEL PLATES CHANNEL WITH MHD

More information

INTRODUCTION TO FLUID MECHANICS June 27, 2013

INTRODUCTION TO FLUID MECHANICS June 27, 2013 INTRODUCTION TO FLUID MECHANICS June 27, 2013 PROBLEM 3 (1 hour) A perfect liquid of constant density ρ and constant viscosity µ fills the space between two infinite parallel walls separated by a distance

More information

10. Buoyancy-driven flow

10. Buoyancy-driven flow 10. Buoyancy-driven flow For such flows to occur, need: Gravity field Variation of density (note: not the same as variable density!) Simplest case: Viscous flow, incompressible fluid, density-variation

More information

Candidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request.

Candidates must show on each answer book the type of calculator used. Log Tables, Statistical Tables and Graph Paper are available on request. UNIVERSITY OF EAST ANGLIA School of Mathematics Spring Semester Examination 2004 FLUID DYNAMICS Time allowed: 3 hours Attempt Question 1 and FOUR other questions. Candidates must show on each answer book

More information

Dimensionless Numbers

Dimensionless Numbers 1 06.10.2017, 09:49 Dimensionless Numbers A. Salih Dept. of Aerospace Engineering IIST, Thiruvananthapuram The nondimensionalization of the governing equations of fluid flow is important for both theoretical

More information

3 Generation and diffusion of vorticity

3 Generation and diffusion of vorticity Version date: March 22, 21 1 3 Generation and diffusion of vorticity 3.1 The vorticity equation We start from Navier Stokes: u t + u u = 1 ρ p + ν 2 u 1) where we have not included a term describing a

More information

Nonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 4,

Nonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 4, Nonlinear Analysis: Modelling and Control, 2008, Vol. 13, No. 4, 513 524 Effects of Temperature Dependent Thermal Conductivity on Magnetohydrodynamic (MHD) Free Convection Flow along a Vertical Flat Plate

More information

UNSTEADY FREE CONVECTION BOUNDARY-LAYER FLOW PAST AN IMPULSIVELY STARTED VERTICAL SURFACE WITH NEWTONIAN HEATING

UNSTEADY FREE CONVECTION BOUNDARY-LAYER FLOW PAST AN IMPULSIVELY STARTED VERTICAL SURFACE WITH NEWTONIAN HEATING FLUID DYNAMICS UNSTEADY FREE CONVECTION BOUNDARY-LAYER FLOW PAST AN IMPULSIVELY STARTED VERTICAL SURFACE WITH NEWTONIAN HEATING R. C. CHAUDHARY, PREETI JAIN Department of Mathematics, University of Rajasthan

More information

7.2 Sublimation. The following assumptions are made in order to solve the problem: Sublimation Over a Flat Plate in a Parallel Flow

7.2 Sublimation. The following assumptions are made in order to solve the problem: Sublimation Over a Flat Plate in a Parallel Flow 7..1 Sublimation Over a Flat Plate in a Parallel Flow The following assumptions are made in order to solve the problem: 1.. 3. The flat plate is very thin and so the thermal resistance along the flat plate

More information

FREE CONVECTIVE HEAT TRANSFER FROM AN OBJECT AT LOW RAYLEIGH NUMBER

FREE CONVECTIVE HEAT TRANSFER FROM AN OBJECT AT LOW RAYLEIGH NUMBER Free Convective Heat Transfer From an Object at Low Rayleigh Number FREE CONVECTIVE HEAT TRANSFER FROM AN OBJECT AT LOW RAYLEIGH NUMBER Md. Golam Kader and Khandkar Aftab Hossain * Department of Mechanical

More information

Shell Balances in Fluid Mechanics

Shell Balances in Fluid Mechanics Shell Balances in Fluid Mechanics R. Shankar Subramanian Department of Chemical and Biomolecular Engineering Clarkson University When fluid flow occurs in a single direction everywhere in a system, shell

More information

Corresponding Author: Kandie K.Joseph. DOI: / Page

Corresponding Author: Kandie K.Joseph. DOI: / Page IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 5 Ver. 1 (Sep. - Oct. 2017), PP 37-47 www.iosrjournals.org Solution of the Non-Linear Third Order Partial Differential

More information

ELEC9712 High Voltage Systems. 1.2 Heat transfer from electrical equipment

ELEC9712 High Voltage Systems. 1.2 Heat transfer from electrical equipment ELEC9712 High Voltage Systems 1.2 Heat transfer from electrical equipment The basic equation governing heat transfer in an item of electrical equipment is the following incremental balance equation, with

More information

Exercise 5: Exact Solutions to the Navier-Stokes Equations I

Exercise 5: Exact Solutions to the Navier-Stokes Equations I Fluid Mechanics, SG4, HT009 September 5, 009 Exercise 5: Exact Solutions to the Navier-Stokes Equations I Example : Plane Couette Flow Consider the flow of a viscous Newtonian fluid between two parallel

More information

Chapter 1. Governing Equations of GFD. 1.1 Mass continuity

Chapter 1. Governing Equations of GFD. 1.1 Mass continuity Chapter 1 Governing Equations of GFD The fluid dynamical governing equations consist of an equation for mass continuity, one for the momentum budget, and one or more additional equations to account for

More information

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell Laminar external natural convection on vertical and horizontal flat plates, over horizontal and vertical cylinders and sphere, as well as plumes, wakes and other types of free flow will be discussed in

More information

Introduction to Heat and Mass Transfer. Week 14

Introduction to Heat and Mass Transfer. Week 14 Introduction to Heat and Mass Transfer Week 14 HW # 7 prob. 2 Hot water at 50C flows through a steel pipe (thermal conductivity 14 W/m-K) of 100 mm outside diameter and 8 mm wall thickness. During winter,

More information

Number of pages in the question paper : 05 Number of questions in the question paper : 48 Modeling Transport Phenomena of Micro-particles Note: Follow the notations used in the lectures. Symbols have their

More information

Chapter 2: Fluid Dynamics Review

Chapter 2: Fluid Dynamics Review 7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading

More information

Joule Heating Effect on the Coupling of Conduction with Magnetohydrodynamic Free Convection Flow from a Vertical Flat Plate

Joule Heating Effect on the Coupling of Conduction with Magnetohydrodynamic Free Convection Flow from a Vertical Flat Plate Nonlinear Analysis: Modelling and Control, 27, Vol. 12, No. 3, 37 316 Joule Heating Effect on the Coupling of Conduction with Magnetohydrodynamic Free Convection Flow from a Vertical Flat Plate M. A. Alim

More information

2. FLUID-FLOW EQUATIONS SPRING 2019

2. FLUID-FLOW EQUATIONS SPRING 2019 2. FLUID-FLOW EQUATIONS SPRING 2019 2.1 Introduction 2.2 Conservative differential equations 2.3 Non-conservative differential equations 2.4 Non-dimensionalisation Summary Examples 2.1 Introduction Fluid

More information

Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition

Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer Contents Preface v 1 Introduction to Kinematics 1 1.1 Fluids and solids 1 1.2 Fluid parcels and flow

More information

Boundary-Layer Theory

Boundary-Layer Theory Hermann Schlichting Klaus Gersten Boundary-Layer Theory With contributions from Egon Krause and Herbert Oertel Jr. Translated by Katherine Mayes 8th Revised and Enlarged Edition With 287 Figures and 22

More information

Review of fluid dynamics

Review of fluid dynamics Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume is a tagged region that moves with the fluid. Hence

More information

FINITE ELEMENT ANALYSIS OF MIXED CONVECTION HEAT TRANSFER ENHANCEMENT OF A HEATED SQUARE HOLLOW CYLINDER IN A LID-DRIVEN RECTANGULAR ENCLOSURE

FINITE ELEMENT ANALYSIS OF MIXED CONVECTION HEAT TRANSFER ENHANCEMENT OF A HEATED SQUARE HOLLOW CYLINDER IN A LID-DRIVEN RECTANGULAR ENCLOSURE Proceedings of the International Conference on Mechanical Engineering 2011 (ICME2011) 18-20 December 2011, Dhaka, Bangladesh ICME11-TH-014 FINITE ELEMENT ANALYSIS OF MIXED CONVECTION HEAT TRANSFER ENHANCEMENT

More information

Fluid Mechanics Qualifying Examination Sample Exam 2

Fluid Mechanics Qualifying Examination Sample Exam 2 Fluid Mechanics Qualifying Examination Sample Exam 2 Allotted Time: 3 Hours The exam is closed book and closed notes. Students are allowed one (double-sided) formula sheet. There are five questions on

More information

PHYSICAL MECHANISM OF NATURAL CONVECTION

PHYSICAL MECHANISM OF NATURAL CONVECTION 1 NATURAL CONVECTION In this chapter, we consider natural convection, where any fluid motion occurs by natural means such as buoyancy. The fluid motion in forced convection is quite noticeable, since a

More information

Differential relations for fluid flow

Differential relations for fluid flow Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow

More information

CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION

CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,

More information

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte

More information

NUMERICAL SOLUTION OF MHD FLOW OVER A MOVING VERTICAL POROUS PLATE WITH HEAT AND MASS TRANSFER

NUMERICAL SOLUTION OF MHD FLOW OVER A MOVING VERTICAL POROUS PLATE WITH HEAT AND MASS TRANSFER Int. J. Chem. Sci.: 1(4), 14, 1487-1499 ISSN 97-768X www.sadgurupublications.com NUMERICAL SOLUTION OF MHD FLOW OVER A MOVING VERTICAL POROUS PLATE WITH HEAT AND MASS TRANSFER R. LAKSHMI a, K. JAYARAMI

More information

INSTRUCTOR: PM DR MAZLAN ABDUL WAHID

INSTRUCTOR: PM DR MAZLAN ABDUL WAHID SMJ 4463: HEAT TRANSFER INSTRUCTOR: PM ABDUL WAHID http://www.fkm.utm.my/~mazlan TEXT: Introduction to Heat Transfer by Incropera, DeWitt, Bergman, Lavine 5 th Edition, John Wiley and Sons Chapter 9 Natural

More information

Rational derivation of the Boussinesq approximation

Rational derivation of the Boussinesq approximation Rational derivation of the Boussinesq approximation Kiyoshi Maruyama Department of Earth and Ocean Sciences, National Defense Academy, Yokosuka, Kanagawa 239-8686, Japan February 22, 2019 Abstract This

More information

CONVECTIVE HEAT TRANSFER

CONVECTIVE HEAT TRANSFER CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran CHAPTER 5 NATURAL CONVECTION HEAT TRANSFER BASIC CONCEPTS MECHANISM OF NATURAL

More information

FLUID MECHANICS. Chapter 9 Flow over Immersed Bodies

FLUID MECHANICS. Chapter 9 Flow over Immersed Bodies FLUID MECHANICS Chapter 9 Flow over Immersed Bodies CHAP 9. FLOW OVER IMMERSED BODIES CONTENTS 9.1 General External Flow Characteristics 9.3 Drag 9.4 Lift 9.1 General External Flow Characteristics 9.1.1

More information

5th WSEAS Int. Conf. on Heat and Mass transfer (HMT'08), Acapulco, Mexico, January 25-27, 2008

5th WSEAS Int. Conf. on Heat and Mass transfer (HMT'08), Acapulco, Mexico, January 25-27, 2008 Numerical Determination of Temperature and Velocity Profiles for Forced and Mixed Convection Flow through Narrow Vertical Rectangular Channels ABDALLA S. HANAFI Mechanical power department Cairo university

More information

MIXED CONVECTION SLIP FLOW WITH TEMPERATURE JUMP ALONG A MOVING PLATE IN PRESENCE OF FREE STREAM

MIXED CONVECTION SLIP FLOW WITH TEMPERATURE JUMP ALONG A MOVING PLATE IN PRESENCE OF FREE STREAM THERMAL SCIENCE, Year 015, Vol. 19, No. 1, pp. 119-18 119 MIXED CONVECTION SLIP FLOW WITH TEMPERATURE JUMP ALONG A MOVING PLATE IN PRESENCE OF FREE STREAM by Gurminder SINGH *a and Oluwole Daniel MAKINDE

More information

The University of the West Indies, St. Augustine, Trinidad and Tobago. The University of the West Indies, St. Augustine, Trinidad and Tobago

The University of the West Indies, St. Augustine, Trinidad and Tobago. The University of the West Indies, St. Augustine, Trinidad and Tobago Unsteady MHD Free Convection Couette Flow Through a Vertical Channel in the Presence of Thermal Radiation With Viscous and Joule Dissipation Effects Using Galerkin's Finite Element Method Victor M. Job

More information

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium Transport in Porous Media (2006) 64: 1 14 Springer 2006 DOI 10.1007/s11242-005-1126-6 Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid past a Vertical Plate Immersed in a Porous Medium

More information

PHYSICAL MECHANISM OF CONVECTION

PHYSICAL MECHANISM OF CONVECTION Tue 8:54:24 AM Slide Nr. 0 of 33 Slides PHYSICAL MECHANISM OF CONVECTION Heat transfer through a fluid is by convection in the presence of bulk fluid motion and by conduction in the absence of it. Chapter

More information

Outlines. simple relations of fluid dynamics Boundary layer analysis. Important for basic understanding of convection heat transfer

Outlines. simple relations of fluid dynamics Boundary layer analysis. Important for basic understanding of convection heat transfer Forced Convection Outlines To examine the methods of calculating convection heat transfer (particularly, the ways of predicting the value of convection heat transfer coefficient, h) Convection heat transfer

More information

Laminar Forced Convection Heat Transfer from Two Heated Square Cylinders in a Bingham Plastic Fluid

Laminar Forced Convection Heat Transfer from Two Heated Square Cylinders in a Bingham Plastic Fluid Laminar Forced Convection Heat Transfer from Two Heated Square Cylinders in a Bingham Plastic Fluid E. Tejaswini 1*, B. Sreenivasulu 2, B. Srinivas 3 1,2,3 Gayatri Vidya Parishad College of Engineering

More information

TABLE OF CONTENTS CHAPTER TITLE PAGE

TABLE OF CONTENTS CHAPTER TITLE PAGE v TABLE OF CONTENTS CHAPTER TITLE PAGE TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS LIST OF APPENDICES v viii ix xii xiv CHAPTER 1 INTRODUCTION 1.1 Introduction 1 1.2 Literature Review

More information

Laminar Mixed Convection in the Entrance Region of Horizontal Quarter Circle Ducts

Laminar Mixed Convection in the Entrance Region of Horizontal Quarter Circle Ducts Proceedings of the 5th IASME/WSEAS Int. Conference on Heat Transfer Thermal Engineering and Environment Athens Greece August 5-7 007 49 Laminar Mixed Convection in the Entrance Region of Horizontal Quarter

More information

Lecture 30 Review of Fluid Flow and Heat Transfer

Lecture 30 Review of Fluid Flow and Heat Transfer Objectives In this lecture you will learn the following We shall summarise the principles used in fluid mechanics and heat transfer. It is assumed that the student has already been exposed to courses in

More information

Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017

Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017 Chemical and Biomolecular Engineering 150A Transport Processes Spring Semester 2017 Objective: Text: To introduce the basic concepts of fluid mechanics and heat transfer necessary for solution of engineering

More information

Laplace Technique on Magnetohydrodynamic Radiating and Chemically Reacting Fluid over an Infinite Vertical Surface

Laplace Technique on Magnetohydrodynamic Radiating and Chemically Reacting Fluid over an Infinite Vertical Surface International Journal of Engineering and Technology Volume 2 No. 4, April, 2012 Laplace Technique on Magnetohydrodynamic Radiating and Chemically Reacting Fluid over an Infinite Vertical Surface 1 Sahin

More information

Lectures on Applied Reactor Technology and Nuclear Power Safety. Lecture No 6

Lectures on Applied Reactor Technology and Nuclear Power Safety. Lecture No 6 Lectures on Nuclear Power Safety Lecture No 6 Title: Introduction to Thermal-Hydraulic Analysis of Nuclear Reactor Cores Department of Energy Technology KTH Spring 2005 Slide No 1 Outline of the Lecture

More information

ENGR Heat Transfer II

ENGR Heat Transfer II ENGR 7901 - Heat Transfer II External Flows 1 Introduction In this chapter we will consider several fundamental flows, namely: the flat plate, the cylinder, the sphere, several other body shapes, and banks

More information

Candidates must show on each answer book the type of calculator used. Only calculators permitted under UEA Regulations may be used.

Candidates must show on each answer book the type of calculator used. Only calculators permitted under UEA Regulations may be used. UNIVERSITY OF EAST ANGLIA School of Mathematics May/June UG Examination 2011 2012 FLUID DYNAMICS MTH-3D41 Time allowed: 3 hours Attempt FIVE questions. Candidates must show on each answer book the type

More information

Contents. Microfluidics - Jens Ducrée Physics: Laminar and Turbulent Flow 1

Contents. Microfluidics - Jens Ducrée Physics: Laminar and Turbulent Flow 1 Contents 1. Introduction 2. Fluids 3. Physics of Microfluidic Systems 4. Microfabrication Technologies 5. Flow Control 6. Micropumps 7. Sensors 8. Ink-Jet Technology 9. Liquid Handling 10.Microarrays 11.Microreactors

More information

Fluid Mechanics II Viscosity and shear stresses

Fluid Mechanics II Viscosity and shear stresses Fluid Mechanics II Viscosity and shear stresses Shear stresses in a Newtonian fluid A fluid at rest can not resist shearing forces. Under the action of such forces it deforms continuously, however small

More information

Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015

Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 I. Introduction (Chapters 1 and 2) A. What is Fluid Mechanics? 1. What is a fluid? 2. What is mechanics? B. Classification of Fluid Flows 1. Viscous

More information

F11AE1 1. C = ρν r r. r u z r

F11AE1 1. C = ρν r r. r u z r F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular cross-section of radius a, whose centre line is aligned along the z-axis; see Figure 1. Assume no-slip boundary conditions

More information

Table of Contents. Foreword... xiii. Preface... xv

Table of Contents. Foreword... xiii. Preface... xv Table of Contents Foreword.... xiii Preface... xv Chapter 1. Fundamental Equations, Dimensionless Numbers... 1 1.1. Fundamental equations... 1 1.1.1. Local equations... 1 1.1.2. Integral conservation equations...

More information

AA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow

AA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit

More information

CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer

CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer CENG 501 Examination Problem: Estimation of Viscosity with a Falling - Cylinder Viscometer You are assigned to design a fallingcylinder viscometer to measure the viscosity of Newtonian liquids. A schematic

More information

UNIVERSITY of LIMERICK

UNIVERSITY of LIMERICK UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF

More information

G. C. Hazarika 2 Department of Mathematics Dibrugarh University, Dibrugarh

G. C. Hazarika 2 Department of Mathematics Dibrugarh University, Dibrugarh Effects of Variable Viscosity and Thermal Conductivity on Heat and Mass Transfer Flow of Micropolar Fluid along a Vertical Plate in Presence of Magnetic Field Parash Moni Thakur 1 Department of Mathematics

More information

Introduction to Heat and Mass Transfer. Week 10

Introduction to Heat and Mass Transfer. Week 10 Introduction to Heat and Mass Transfer Week 10 Concentration Boundary Layer No concentration jump condition requires species adjacent to surface to have same concentration as at the surface Owing to concentration

More information

Lesson 6 Review of fundamentals: Fluid flow

Lesson 6 Review of fundamentals: Fluid flow Lesson 6 Review of fundamentals: Fluid flow The specific objective of this lesson is to conduct a brief review of the fundamentals of fluid flow and present: A general equation for conservation of mass

More information

COMBINED EFFECTS OF RADIATION AND JOULE HEATING WITH VISCOUS DISSIPATION ON MAGNETOHYDRODYNAMIC FREE CONVECTION FLOW AROUND A SPHERE

COMBINED EFFECTS OF RADIATION AND JOULE HEATING WITH VISCOUS DISSIPATION ON MAGNETOHYDRODYNAMIC FREE CONVECTION FLOW AROUND A SPHERE Suranaree J. Sci. Technol. Vol. 20 No. 4; October - December 2013 257 COMBINED EFFECTS OF RADIATION AND JOULE HEATING WITH VISCOUS DISSIPATION ON MAGNETOHYDRODYNAMIC FREE CONVECTION FLOW AROUND A SPHERE

More information

COURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour. Basic Equations in fluid Dynamics

COURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour. Basic Equations in fluid Dynamics COURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour Basic Equations in fluid Dynamics Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1 Description of Fluid

More information

Contents. I Introduction 1. Preface. xiii

Contents. I Introduction 1. Preface. xiii Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics.........................

More information

Getting started: CFD notation

Getting started: CFD notation PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =

More information

THE UNSTEADY FREE CONVECTION FLOW OF ROTATING MHD SECOND GRADE FLUID IN POROUS MEDIUM WITH EFFECT OF RAMPED WALL TEMPERATURE

THE UNSTEADY FREE CONVECTION FLOW OF ROTATING MHD SECOND GRADE FLUID IN POROUS MEDIUM WITH EFFECT OF RAMPED WALL TEMPERATURE THE UNSTEADY FREE CONVECTION FLOW OF ROTATING MHD SECOND GRADE FLUID IN POROUS MEDIUM WITH EFFECT OF RAMPED WALL TEMPERATURE 1 AHMAD QUSHAIRI MOHAMAD, ILYAS KHAN, 3 ZULKHIBRI ISMAIL AND 4* SHARIDAN SHAFIE

More information

Convective Mass Transfer

Convective Mass Transfer Convective Mass Transfer Definition of convective mass transfer: The transport of material between a boundary surface and a moving fluid or between two immiscible moving fluids separated by a mobile interface

More information

Flow and Natural Convection Heat Transfer in a Power Law Fluid Past a Vertical Plate with Heat Generation

Flow and Natural Convection Heat Transfer in a Power Law Fluid Past a Vertical Plate with Heat Generation ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.7(2009) No.1,pp.50-56 Flow and Natural Convection Heat Transfer in a Power Law Fluid Past a Vertical Plate with

More information