Ignore Gdiff,j,x as a boundary layer approximation and substitute into the continuity equation. Assume steady flow, steady state, and constant properties. For the momentum equation neglect body forces. For the energy equation neglect body force work and the energy source. Also, for the energy equation use the thermal equation of state for ideal gases or incompressible liquids, and assume the Mach number is small for the case of the ideal gas, but do not neglect viscous dissipation.
Assume boundary-layer flow, but do not neglect axial conduction. For constant properties, Eq. Be sure to state all assumptions made. These steps are basically the redevelopment of the formulation leading up to Eq. Note that Eq. Neglect the kinetic energy term, and note that there is no wall transpiration.
Let mj,s be the mass concentration of j at the wall. Continuity Eq. These steps are essentially a repeat of the formulation leading up to Eq. Note that Eqs. Their forms differ in the convection term and in the stress or diffusion term. The form of the convective term in is called the conservative form. Chain-rule differentiate the convective term of and subtract from it to modify the convection term, then assume constant density for the time term. This recovers the left hand side of Now, substitute the stress tensor, into and substitute for the strain-rate tensor.
Assume constant viscosity and constant density. For constant density flow, the second coefficient of viscosity the dilatation term is eliminated. The second term in the strain rate tensor also disappears from the conservation of mass assuming constant density , and the result is the right hand side of Multiply Eq. Note that you will first have to chain-rule the viscous work term in to obtain the unsteady static enthalpy equation.
Now, substitute along with the strain-rate tensor into the static enthalpy equation. The result is We combine the definition of stagnation enthalpy following Eq. The result is , similar to what is found in Ref.
The first step is to chain-rule the conservative form of the convective term that appears in Eq. The pressure gradient becomes an ordinary derivative. This also applies to the Reynolds stress term. Note Eq. Recast the laminar boundary-layer equations for momentum, Eq.
The structure of the equations follow the format of Eq. An equivalent energy source term would be a viscous dissipation term, which could be added. Let the velocity be uniform at the duct entrance. Calculate the development of the velocity profile in the entry length, using the momentum integral equation , and an assumption that the velocity profile may be approximated by a constant-velocity segment across the center portion of the duct and by simple parabolas in the growing boundary layer adjacent to the surfaces.
Note that the mean velocity, Eq. Three boundary conditions are needed to determine a, b, c. The other two come by applying boundary conditions at the edge of the developing boundary layer within the channel, i.
The wall shear stress must also be evaluated, following the idea of Eq. Start with a control volume that is of infinitesimal dimension in the flow direction but that extends across the entire flow section. Then reconsider the problem when fluid density varies in some known manner along the tube but can be considered as effectively constant over the flow cross section. Discuss the implications of the latter assumption. This problem can be solved two ways.
The first method involves partially integrating the differential momentum equation over the flow cross-section. Reformulate Eq. Note the similarity to Eq. Rearranging, assuming constant density, and dividing through by the dynamic pressure term yields Eq. Note the introduction of the mean friction coefficient from Eq. Note that a change in density will lead to a change in Reynolds number. At a particular flow cross section, calculate the total axial momentum flux by integration over the entire cross section.
Compare this with the momentum flux evaluated by multiplying the mass flow rate times the mean velocity.
Explain the difference, then discuss the implications for the last part of Prob. For fully-developed flow in a circular tube, the velocity profile is given by Eq. The tubes are 2 m long. One of the tanks has a higher pressure than the other, and air flows through the two tubes at a combined rate of 0. Assuming that fluid properties remain constant and that the entrance and exit pressure losses are negligible, calculate the pressure differences between the two tanks.
For one of the fluids the plate separation is 1 cm and the nominal fin separation is 2 mm. However, manufacturing tolerance uncertainties lead to the possibility of a 10 percent variation in the fin separation. Consider the extreme case where a 10 percent oversize passage is adjacent to a 10 percent undersize passage.
Let the flow be laminar and the passages sufficiently long that an assumption of fully developed flow throughout is reasonable. For a fixed pressure drop, how does the flow rate differ for these two passages and how does it compare to what it would be if the tolerance were zero? Note, this problem requires D to be replaced by Dh, the hydraulic diameter. Note also, we can use Eq. For this problem, the assumption of fully-developed flow is made, and Figure 7- 4 can be used to determine c f Re for these rectangular heat exchanger passages.
Define a control volume, such as in Fig. Note that there will be no momentum flux difference into and out of the control volume because of the fully-developed assumption. Applying the momentum theorem Eq. Forming the ratio of Eq. This is a very important result for internal flows, because it applies to both laminar and turbulent pipe flow.
An alternative solution is to derive the velocity profile for pipe flow, Eq. Then forming the ratio of these shear stresses yields Eq.
Compare your friction result with Fig. For the parallel-planes geometry y is measured from the centerline and the channel is of width a. Assume fully-developed laminar flow with constant properties. Next, we create the mean velocity, following Eq. Again, note the similarity to Eq. Now follow the procedure of Eq. Comparing the result to Fig. In Fig. Note this is the geometry of journal bearings. Compare your velocity-profile result with Eq. For flow in the annular space between two concentric pipes, there will be a peak in the velocity profile, and it is located at rm compared to the centerline for the pipe and parallel- planes channel.
Again, note the approximate similarity to Eq. Note that x D Re is the inverse of the Langhaar variable, used in Fig. Use constant fluid properties, and note that the energy equation does not have to be solved. For initial conditions let the velocity profile be flat at a value equal to the mean velocity. Use Eq. Plot the nondimensional velocity profiles at various x Dh Re D locations h and compare to Fig. Plot the absolute value of the pressure gradient versus x Dh Re D to show how the gradient becomes constant beyond the hydrodynamic entrance region.
Note a small correction to the problem write-up. The data file for this problem is 7. Only the momentum results will be discussed. Here is an abbreviated listing of the output file it will be called out. The file ftn For a more accurate integration of the local friction coefficient to obtain its mean value, you will want to increase the number of data points in ftn If you work with the friction coefficient rather than the wall shear stress, be careful, because cf2 is the friction coefficient divided by two.
The following graph is the reduced data for the 3 friction coefficients, plotted versus x Dh Re D rather than using the Langhaar variable. Because you have the h pressure drop data versus x you can easily construct the pressure gradient variation with x and plot its absolute value logarithmically versus x Dh Re D a ln-ln plot to see where the gradient becomes h constant.
This is a measure of the entry region. The output profile will then contain velocity profiles at each x m station. The profiles will have a set of shapes matching Fig. This ratio is also a measure of fully-developed flow. Note, for the parallel-planes geometry the variable rw m is the half-width of the channel. The cfapp value requires h nearly four times more length for its asymptotic state to be reached. This difference can be seen in the references by Shah and London3 and Shah.
For example, a value of 0. This is reflective of the viscous transport mechanism in laminar flows where the core flow adjustment is not in sync with the wall shear flow region. The profiles will have a set of shapes similar to that for a pipe in Fig. Plot the absolute value of the pressure gradient versus x Dh Re D to show how h the gradient becomes constant beyond the hydrodynamic entrance region. Evaluate the ratio of centerline velocity to mean velocity and plot it versus x Dh Re D to show how the ratio becomes a h constant beyond the hydrodynamic entrance region.
Note several small corrections to the problem write-up. The x Dh Re D variable is the reciprocal of the Langhaar variable.
Plot the entry-region h distributions of the local friction coefficients for inner and outer radius, the area-weighted average friction coefficient, and the apparent friction coefficients. The idea of evaluating the centerline velocity is not meaningful for an annulus.
Instead, examine the maximum velocity and the location of the maximum. The analysis leading to these formulations is given as a part of problem and it can be found in Shah and London.
The c f m x value is not generally used for the annulus. For a more accurate pressure gradient, you will want to increase the number of data points in ftn Note also that cf2 is the friction coefficient divided by two. Because you have the pressure drop data versus x you can easily construct the pressure gradient variation with x and plot its absolute value logarithmically versus x Dh Re D a ln-ln plot to h see where the gradient becomes constant.
The student can find the maximum velocity within the annulus and plot this ratio as a measure of fully-developed flow. However, for this radius ratio the profile shape and maximum is very similar to that for the parallel planes. Problem 2. The heat-transfer rate per unit of duct length is constant. Compare your results with those given in the text Table Let the plate spacing be a.
Assume steady laminar flow with constant properties. For this coordinate system, the velocity profile has been derived in problem That is, only C1 can be resolved.
Note, the solution for this problem can also be carried out without strictly considering linear superposition. The data set construction is based on the s Note this is not the correct thermal initial condition for this problem but the solution will converge to a thermally fully-developed solution.
Note a slight change in the problem statement. Here is an abbreviated listing of the 8. The files ftn Here is an abbreviated output from ftn For the E-surface, from the ftn To better understand any negative Nusselt number, the temperature profiles need to be examined. The following plot compares the profiles for both heating ratios.
Of equal importance is the examination of the temperature profiles. Thus, as the Prandtl number approaches zero, the temperature profile can approach a fully developed form before the velocity profile has even started to develop although this is a situation of purely academic interest. Convection solutions based on a uniform velocity over the cross section, as described, are called slug-flow solutions.
Develop an expression for the slug-flow, fully developed temperature-profile Nusselt number for constant heat rate per unit of tube length for a concentric circular-tube annulus with a radius ratio of 0.
Compare with the results in Table and discuss. Now consider the energy equation This equation assumes no viscous dissipation, and it assumes constant properties for an ideal gas and steady state. Let the temperature profile vary with x and r only. These two boundary conditions are very much similar to those for a circular pipe zero heat flux at the pipe centerline and a uniform heat flux at the surface. Thus the boundary conditions are to that following Eq. Now compute the mean temperature using Eq.
The first way is form an energy balance similar to Eq. The profile shapes are not that dissimilar. At some point beyond the fully developed location, a 1. Let the mass flow rate of the fuel be 1. Calculate and plot both tube surface temperature and fluid mean temperature as functions of tube length. What is the highest temperature experienced by any of the fluid?. The variables that were changed in s Note: feel free to adjust the selected x m values.
Suppose heat is transferred to the fluid on one side and out of the fluid on the other at the same rate. What is the Nusselt number on each side of the passage? Sketch the temperature profile. Suppose the fluid is an oil for which the viscosity varies greatly with temperature, but all the other properties are relatively unaffected by temperature.
Is the velocity profile affected? Is the temperature profile affected? Is the Nusselt number affected? Note that the Nusselt number functions for this problem have been derived as a part of problem The Nusselt number for the lower surface is Eq. To show that the temperature profile is linear we start with Eq. Note this is not the correct thermal initial condition but it will converge to a thermally fully-developed solution.
This data set is for thermally fully-developed flow between parallel planes with a constant heat flux surface. For this problem make the channel length about hydraulic diameters the problem statement says However, we see it takes quite a long time to reach this thermally fully-developed flow. This is because TEXSTAN does not have a correct thermally fully-developed temperature profile for a flux-flux thermal boundary condition, and the incorrect initial profile takes time to damp out in the output you will see it will take about hydraulic diameters for the solution to converge towards a thermally fully-developed solution, whereas the velocity solution is correct from the xstart location.
Heat is supplied to the inner tube, and the outer tube is externally insulated. The radiation emissivity of both tube surfaces is 0. What is total heat flux from the inner-tube surface at this point? What is the outer-tube surface temperature at this point? What percentage of the heat supplied to the inner tube is transferred directly to the air, and what percentage indirectly from the outer surface?
Assume that the Reynolds number is sufficiently low that the flow is laminar. Assume that the air is transparent to the thermal radiation. Make use of any of the material in the text as needed. First formulate the thermal radiation exchange using a radiative thermal circuit for exchange between two diffuse-gray surfaces with a non-participating medium between the surfaces. This formulation can be found in most undergraduate heat transfer texts.
Now we consider the convective heat transfer for the two surfaces. Note, this value can vary depending on how closely you converge the solution and on the property of air which was selected at K. A particular air-cooled reactor is to be constructed of a stack of fuel plates with a 3-mm air space between them. The length of the flow passage will be 1. The air mass velocity is to be 7.
Prepare a scale plot of heat flux, air mean temperature, and plate surface temperature as a function of distance along the flow passage. Although the heat flux is not constant along the passage, the passage length-to-gap ratio is sufficiently large that the constant-heat-rate heat-transfer solution for the conductance h is not a bad approximation.
Therefore assume h is a constant. We are most interested here in the peak surface temperature; if this occurs in a region where the heat flux is varying only slowly, the approximation is still better.
This is a point for discussion. The solution begins by performing an energy balance on a control volume for an element of the parallel- planes channel, similar to that depicted in Fig.
Let H be the channel height and L be the channel length. We see that the approximation of a thermally fully-developed entry region may not be a good assumption, but it can be verified by TEXSTAN. You must construct the boundary condition table in the input data set to reflect the sine-function surface heat flux distribution.
For internal flows, the numerical mesh is required to extend from surface to centerline or surface to surface. For entry flows, the developing shear layer boundary layer is a very small part of this mesh, and therefore TEXSTAN must take very small flow-direction integration steps. Therefore, internal entry flows require a different stepsize mechanism.
Note the integration stepsize control starts extremely small and then increases, as reflected in aux1 m starting at 0. If we were using cf,app as a measure of hydrodynamically fully developed flow, this number would be about four times larger. Problem 8. If the tube is effectively insulated, calculate and plot the temperature distribution, resulting from frictional heating, in terms of the pertinent parameters.
Start with Eq. For this problem we will assume the properties are constant. This may not be a good assumption, depending on the heat rise due to the viscous dissipation. The Reynolds number for this flow is This is a special case of the constant heat rate problem, and from Eq. Therefore, we can not find the two constants of integration when we separate variables and integrate.
Thus the boundary conditions are the same as those following Eq. Now develop an equation for the mean temperature gradient by performing an energy balance similar to Fig. The wall will be hotter than the core. As the viscous heating increases the overall temperature field, the strong variation of the thermal properties will affect the solution.
Let the journal diameter be 7. Calculate the rate of heat transfer per square meter of bearing surface. Assume no eccentricity, that is, no load on the bearing. How much power is needed to rotate the journal if the bearing is 10 cm long? This analysis can be carried out in either axisymmetric coordinates or in Cartesian coordinates, and the answers will be the same, because the journal clearance is such that radial effects on mass flow rate are negligible. For simplicity we choose the Cartesian analysis.
The bearing forms the outer surface, and it has a constant surface temperature. Define the coordinate y to originate from the inner journal surface, and the clearance distance is H. This problem reduces to a couette- flow problem, and for constant properties, the velocity profile will be linear. The governing equation is Eq. The governing energy equation for constant properties is Eq. The shaft work does work on the fluid, which causes viscous dissipation to overcome friction, which in turn generates heat, and that heat must be removed at the outer surface to maintain steady state couette flow no heat rise in the x-direction, where x is the direction of rotation.
Use variable-surface-temperature theory. This problem concerns the section in the chapter about the effect of axial variation of the surface temperature with hydrodynamically fully developed flow using linear superposition theory.
The surface heat flux is given by Eq. For a step temperature change, the nondimensional temperature profile for the mean temperature is given by Eq. Thereafter let the tube surface be adiabatic. This problem concerns the section in the chapter about the effect of axial variation of the surface heat flux with hydrodynamically fully developed flow using linear superposition theory.
The surface temperature is given by Eq. To evaluate the series you must use at least 20 terms. Table contains the first 5 terms, so use the algorithm at the bottom of the table to generate the 6th through the 20th terms. The reason for the large number of terms is the series is simulating a step heat flux similar to a square wave , rather than a slowly-changing heat flux.
The x m values in the data set are intended to create more integration points in the initial part of the developing thermal boundary layer region based on the logarithmic behavior of the parameter x Dh Re.
We expect complete agreement between TEXSTAN and analysis for the mean temperature because this both procedures incorporate the First Law energy balance into their respective formulations. How does the Nusselt number vary along the heated segments after the effects of the original entry length have damped out?
Note a slight change to the input instructions. For the geometry, choose a tube radius of 1 cm. Because of the variable heating, it is easiest to set all aux1 m values to a uniform integration stepsize of 0. Here is a partial listing of how the variable heat flux boundary condition has been set up in the 8. Periodic heating can be a very effective means of achieving a high heat transfer coefficient.
However, the overall temperature rise is still governed by the energy balance. The graph shown below compares the mean and surface temperatures predicted by TEXSTAN with the analysis for 3 intervals using 20 terms in the series note more terms will considerably improve the comparison.
Note the problem statement should include the requirement of a constant surface temperature boundary condition. The analysis uses Eq. Explain physically the reasons for the behavior noted. Beta is 6. For this problem the surface temperature is given by Eq. Let there be heat transfer from the inner tube only outer tube insulated , and let the heat flux on the inner tube vary as in Prob.
Describe in detail a computing procedure for evaluating both the inner- and outer- tube surface temperatures as functions of length along the tube. Assume that the heat-transfer resistance of the wall is negligible in the radial direction and that fluid properties are constant. This problem is designed as an application of Eq. To show that plane radiation incident on a circular tube is a cosine function, compute the radiation view factor or shape factor or configuration factor between the plane and the cylinder, and you will show it contains a cosine.
The Nusselt number variation is given by Eq. The re-radiation will produce a heat flux from the tube that subtracts slightly from the inflow on one side, and results in net radiation from the tube on the other. Let the fluid temperature at the tube entrance be uniform at Te. What are the implications of this result? Discuss how the constant b affects the asymptotic Nusselt number. This problem is solved by substituting the proposed expression for Ts into Eq.
Let there be heat transfer to or from the fluid at a constant rate per unit of tube length. Determine the Nusselt number if the effect of frictional heating viscous mechanical energy dissipation is included in the analysis. How does frictional heating affect the Nusselt number? What are the significant new parameters? Consider some numerical examples and discuss the results. For the tube geometry r is measured from the centerline of the inner pipe and rs is the radius of the tube.
Add in a viscous dissipation term similar to Eq. Determine an expression for the Nusselt number as a function of the pertinent parameters. What are they? Evaluate the convection conductance in the usual manner, on the basis of heat flux through the surface, surface temperature, and fluid mixed mean temperature.
Add in a volumetric source term similar to Eq. From an examination of the applicable energy differential equation alone, deduce the approximate shape of the temperature profile within the fluid, and determine whether the highest temperature of the fluid at any axial position occurs at the tube surface or at the tube centerline. Explain the reasons for the result. This solution is a variation on problem The mean temperature continues to increase with x. The lowest temperature is at the centerline, and the maximum is at the wall.
Let the Prandtl number be 0. Compare the results with Table Let the Reynolds number be , and pick fluid properties that are appropriate to the chosen Prandtl number. Use constant fluid properties and do not consider viscous dissipation.
For initial conditions let the velocity profile be hydrodynamically fully developed and the temperature profile be flat at some value Te. The data file for this problem is 8.
The file out. For this problem, choose Pr values of 0. There are three data files for this, one for each Prandtl number. Each data set is labeled 8. The x m array has to be adjusted for each Pr value. The thermal entry region correlates logarithmically with 2 x Dh Re Dh Pr , so this becomes an easy way of estimating how to create a distribution of x- locations for varying aux1. Note the Nu values are not calculated as the heat transfer approaches zero. Because this problem has symmetrical thermal boundary conditions, choose the option in TEXSTAN that permits the centerline of the parallel planes channel to be a symmetry line.
This matches the conclusion in problem for the hydrodynamic entry region. As with the hydrodynamic problem, it is important to remember that there are two contributions to the total pressure drop, surface friction and flow acceleration. The combined effect leads to a longer entry length than is measured by the wall friction. Problem 50 40 Nu x 30 Nu x 20 10 0 0.
Because this problem has asymmetrical thermal boundary conditions, choose the option in TEXSTAN that permits the calculation from surface to surface of the parallel-plane channel. For initial conditions let the velocity profile be flat at a value equal to the mean velocity and the temperature profile be flat at some value Te. Compare the results with the Nu11 values for parallel planes in Table note the influence coefficient is not used for this problem.
For initial conditions let the velocity profile be flat at a value equal to the mean velocity and the temperature profile be flat at some value te. Because this problem has asymmetrical thermal boundary conditions, choose the option in TEXSTAN that permits the calculation from surface to surface.
Test two cases: constant heat flux on the inside surface and the outside surface adiabatic, and then the opposite case. Calculate the flow and compare the results Fig. Discuss the behavior of the various variables in terms of the temperature profiles obtained as a part of the computer analysis. The Ts x distribution is linear over the interval 0. The deltax control was handled similarly to problem ,. A smaller value would generate more points. Here is an abbreviated listing of the output file ftn A more detailed distribution is shown in the figure below.
Problem Tm Ts Temperature 0. The heat transfer information are plotted in the figure below. To fully understand the variable surface temperature behavior it is important that you plot the temperature profiles. Calculate the displacement thickness of the boundary layer at the stagnation point, and discuss the significance of the result.
For this problem we will assume a constant density, i. Density variation throughout the boundary layer is caused either by an imposed thermal boundary condition wall temperature or wall heat flux which leads to a large wall-to- free stream temperature difference, or when viscous heating associated with viscous work cause local temperature variation in the region near the surface.
For gases, we find experimentally that we can ignore variable properties including density when 0. For this problem we have no information about the thermal boundary condition.
Thus, displacement thickness Eq. One source is tabulated in Schlichting1. The other source is to solve Eq. An alternative solution for this problem is to use Eq.
Evaluation yields a value of d2 of about 0. This permits the parameter defined by Eq. In complex variable theory, potential flow solutions are greatly simplified if the flow domain can be transformed into a semi- infinite domain. The domain is now transformed to the upper-half of the w-plane. The applicable momentum integral equation is , rewritten as Eq. From Eq. Thus, it is easily added to Eq.
Calculate the air mass flow rate through the nozzle and the overall pressure drop through the nozzle. If you were to define a Reynolds number based on throat diameter and mean velocity, how would the discharge coefficient vary with Reynolds number? The analysis procedure for this problem is: to calculate d2 using Eq. The problem requires assumptions of steady flow and constant properties. There are several geometric variables to be defined.
Equation can be evaluated using standard methods for the integral. Interpolate Table for the shape factor H, and then compute the displacement thickness. The data file for this problem is 9. Because it is axisymmetric flow, the geometry transverse radius variable of the nozzle wall, R x in this analysis, is used to create the array rw m at x m locations. Plotting this pressure-gradient distribution will help the user be sure their free stream velocity array, ubI m , is differentially smooth.
Here is an abbreviated listing of the file ftn Upon introducing the functional form of G, the equation separates and Eq. A second procedure is to transform the differential operators and y simultaneously. Either works. The geometrical dimensions of the plate are 1 m wide a unit width by 0. Calculate the boundary layer flow and compare the results with the similarity solution for development in the streamwise direction of such quantities as the boundary-layer thickness see Table , displacement thickness see Eq.
Evaluate the concept of boundary-layer similarity by comparing non- dimensional velocity profiles at several x-locations to themselves and to Table Compare the friction coefficient results based on x-Reynolds number with Eq. Calculate the friction coefficient distribution using momentum integral Eq. Feel free to investigate any other attribute of the boundary-layer flow.
Here are abbreviated listings of these files. We can use these ratios to help determine if a data set construction is correct. The profiles will be printed as a part of the file out. You can choose where to print the profiles by adding x locations to the x m. Be sure to change the two nxbc variables and add the appropriate sets of two lines of boundary condition information for each new x- location.
This is explained in detail in the s Problem 1. Note that the profile comes from solving Eq. Calculate the boundary layer flow and compare the friction coefficient results based on x- Reynolds number with the results in the text. Evaluate the concept of boundary-layer similarity by comparing nondimensional velocity profiles at several x-locations with each other. Feel free to investigate any other attribute of the boundary-layer flow and to compare your results with other open-literature solutions.
There needs to be a slight modification to the instructions in the problem statement regarding the calculations of xstart and axx. Note the much thinner boundary layer thickness.
See App. C for tables of error and gamma functions. This problem is the Blasius solution to the flat plate boundary layer with constant free stream velocity and constant surface temperature.
Substituting the linear representation into Eq. Compare with the approximate results of Prob. Note that numerical integration is required. Here is the formulation necessary for the Runge-Kutta method. However, we really need to use much larger infinite-state values to insure the boundary value problem is correct, and nothing is lost by using a much larger number.
This can be compared to Eq. The second method for solution of this problem is to use the similarity analysis on p. From this, develop an equation for heat transfer at the stagnation point of a circular cylinder in cross flow, in terms of the oncoming velocity and the diameter of the tube.
The governing equation for momentum with the Falkner-Skan free-stream velocity distribution is given by Eq. Show that a similarity solution to the energy equation is obtainable under these conditions. This problem is a Blasius type solution to the flat plate boundary layer with constant free stream velocity and variable surface temperature.
Compare with the exact result from Prob. The solution procedure is to evaluate the surface heat flux for the given surface temperature distribution and then formulate the Nusselt number. Combine Eqs. Using the results of Prob. The analysis in Prob. For the shear thickness, combine Eq. Let the cylinder be of a thin-walled porous material so that air can be pumped inside the cylinder and out through the pores in order to cool the walls.
The objective of the problem is to calculate the cylinder surface temperature in the region of the stagnation point for various cooling-air flow rates, expressed as the mass rate of cooling air per square meter of cylinder surface. The same cooling air could be used to cool the surface internally by convection without passing through the surface out into the main stream. The first task is to perform an energy balance on a unit of surface area in the stagnation region.
Let Ts be the wall, Ti be the injection or internal coolant temperature. Let the plate be divided into three sections, each 10 cm in flow length. Evaluate and plot the heat flux at all points along the 30 cm of plate length, and find the local heat-transfer coefficient.
Choose a starting x-location near the leading edge, say 0. The piecewise surface temperature boundary condition is modeled easily in TEXSTAN by providing temperatures at two x locations for each segment, e. Because TEXSTAN linearly interpolates the surface thermal boundary condition between consecutive x locations, a total of six boundary condition locations is sufficient to describe the surface temperature variation.
To plot the profiles, you will need to add several x m points to the file If the surface temperature were not changing, the thermal boundary layer near the wall will eventually grow outward and the heat flux will become negative.
Problem 0. For the surface temperature distribution, break up the length of the plate into 10 to 20 segments and evaluate the surface temperature at these x locations. These values then become the variable surface temperature boundary condition. Note that a larger number of points will more closely model the sine function. The solution analysis involves evaluation of the surface heat flux for the given surface temperature distribution and then formulate the Nusselt number.
For the analytical approach, convert the integral using the ideas of Appendix C and a Taylor-series approximation for the cosine function. The solutions compare. Note also that the textbook solution relating to Eq. Compare these results with the experimental data for the average Nusselt number around a cylinder. What can you conclude about the heat-transfer behavior in the wake region on the rear surface of the cylinder? For the velocity distribution, break up the surface length of the cylinder over which the boundary layer flows into at least 20 segments and evaluate the velocity at these x-locations.
These values then become the variable velocity boundary condition. A larger number of points will more closely model the distribution, which is especially important because this distribution is differentiated to formulate the pressure gradient, as described by Eq. The solution follows the analysis for flow over a constant-temperature body of arbitrary shape that incorporates the assumption of local flow similarity. The final result is Eq. This gives a cylinder-diameter Reynolds number of , There are several output files that are useful for the cylinder.
The variable R is the radius of revolution, or transverse radius of curvature of the sphere, as shown in Fig. Here is the table, constructed using 15 values at theta angles of 0. Note you can not choose a theta angle of 0 degrees, as this presents a radius singularity x m rw m aux1 m aux2 m aux3 m 0. This gives a sphere- diameter Reynolds number of , Note that the initial condition profiles currently programmed into TEXSTAN for a sphere are approximate profiles, and this is the source of error for small theta.
This will be partly related to not having correct initial conditions. The plate is 15 cm long in the flow direction. The entire surface of the plate is adiabatic except for a 2. Plot the temperature distribution along the entire plate surface.
Discuss the significance of this problem with respect to wing deicing. A tabulation of incomplete beta functions, necessary for this solution, is found in App. The piecewise surface heat flux boundary condition is modeled easily in TEXSTAN by providing heat flux values at two x-locations for each segment, e.
Because TEXSTAN linearly interpolates the surface thermal boundary condition between consecutive x- locations, a total of 6 boundary condition locations is sufficient to describe the surface heat flux variation. The solution procedure is to evaluate the surface temperature distribution for the given surface heat flux distribution. The integral becomes. Problem How does the surface temperature vary for the next 45 cm?
Hint: Note that the first 15 cm must be treated as a surface-temperature-specified problem, while the last 45 cm must be treated as a surface-heat-flux-specified problem. The solution procedure is to evaluate the surface temperature distribution for the variable surface heat flux distribution using Eq.
This is not completely correct. The heat flux formula given above was used to set the heat flux boundary condition, and 8 x m locations were used. This is a minimum number because TEXSTAN linearly interpolates between boundary condition points for heat flux, and the flux is rapidly changing with x. How does this compare with the coefficient far downstream in the tube if the tube surface is at a uniform temperature?
The solution to this problem follows that for problem to some degree. The solution involves calculating the Stanton number for a given Pr using Eq. The free stream velocity just outside the boundary layer of the nozzle wall really needs to be developed from an inviscid or Euler analysis of the incompressible flow in a converging nozzle. The first approximation for this axisymmetric nozzle is to use the model defined by problem The integral can be evaluated using standard methods.
Because acceleration strongly suppresses the momentum and displacement thicknesses, we did not see this effect in problem There is no acceleration term in the corresponding energy equation and the convective transport is changed by the free stream velocity distribution. For each Prandtl number calculate the boundary layer flow and evaluate the concept of boundary-layer similarity by comparing nondimensional temperature profiles at several x locations to themselves for independence of x.
Compare the Nusselt number results based on x-Reynolds number with Eq. Convert the Nusselt number to Stanton number and compare the Stanton number results based on x-Reynolds number with Eq.
Calculate the Stanton number distribution using energy integral Eq. The data file for this problem is For this problem statement, the output file ftn The print variable k5 was changed to reduce the lines of output for this example, but it would typically much lower to obtain a high resolution for plotting.
Using properties at about this temperature, the output file ftn Here is the Stanton number data for comparison with Eq. The plot below compares St Pr 4 3 and the slope will be 0. Be sure to change the two nxbc variables and add the appropriate sets of two lines of boundary condition information for each new x-location.
We can compare this plot with Fig. Note that this equation requires constant surface temperature and constant free stream velocity and flow over a flat surface. The enthalpy thickness distribution is contained in the output file ftn In the world of experimental heat transfer, this can be a good estimation of the Stanton number. The energy boundary conditions are a free stream temperature of K and a constant surface temperature of K. Compare the Nusselt number results based on x-Reynolds number with Table 2.
Convert the Nusselt number to Stanton number calculate the Stanton number distribution using energy integral Eq. Also, evaluate the validity of the approximate integral solution, Eq. It is interesting to examine the output file f Note this accelerating flow causes a fairly strong pressure gradient, and the initial pressure needs to be large enough so the pressure does not go negative.
For this data set it was increased to 4 atmospheres. To confirm the energy integral equation for computing Stanton number, we use the ideas in Chapter 5. Be careful, you can not use Eq. Instead, use Eq. It is presumed that a programmable computer is used for this problem.
Note that you need to develop a new relation for both the viscous sublayer and the fully turbulent region, and the apparent thickness of the sublayer will be a constant to be determined from experiments. Plot these profiles on semi-logarithmic paper and superimpose the equation you have derived for the fully turbulent region, determining the apparent sublayer thickness from the best fit to the data.
This is simply the result of fairly large experimental uncertainty for the experiments with strong suction. Let the boundary layer starting at the origin of the plate be laminar, but assume that a transition to a turbulent boundary layer takes place abruptly at some prescribed critical Reynolds number. Assuming that at the point of transition the momentum thickness of the turbulent boundary layer is the same as the laminar boundary layer and this is a point for discussion , calculate the development of the turbulent boundary layer and the friction coefficient for the turbulent boundary layer.
Plot the friction coefficient as a function of Reynolds number on log—log paper for transition Reynolds numbers based on distance from the leading edge of , and 1,,, and compare with the turbulent flow friction coefficient that would obtain were the boundary layer turbulent from the plate origin. Following the steps for this equation leads to 0. The working fluid is helium, and the stagnation pressure and temperature are kPa and K, respectively.
Assuming one-dimensional isentropic flow, constant specific heats, and a specific heat ratio of 1. Then, assuming that a laminar boundary layer originates at the corner where the convergence starts, calculate the momentum thickness of the boundary layer and the momentum thickness Reynolds number as functions ofdistance along the surface. Assume that a transition to a turbulent boundary layer takes place if and when the momentum thickness Reynolds number exceeds An approximate analysis may be carried out on the assumption of constant fluid properties, in which case let the properties be those obtaining at the throat.
Alternatively, a better approximation can be based on the results of Prob. Calculate the displacement thickness of the boundary layer at the throat of the nozzle. Is any correction to the mass flow rate warranted on the basis of this latter calculation? Assume that a transition to a turbulent boundary layer takes place when the momentum thickness Reynolds number exceeds Describe how the forces acting on the blade could be analyzed from the given data. This problem is a direct application of Eq.
The R cancels out in both cases. Problems and should be solved first to get equation forms applicable to varying free-stream density.
The laminar stagnation region near the leading edge is probably best handled by using Eq. The momentum thickness at the stagnation point will be finite, not zero.
Because there is a laminar boundary layer preceding the turbulent boundary layer, Eq. The distance y is measured from the plane of the tops of the balls. The objective of this problem is to analyze these data in the framework of the rough-surface theory developed in the text. What is the apparent value of ks? Of Rek? What is the roughness regime? How does the wake compare with that of a smooth surface?
Do the data support the theory? The use of the plane of the tops of the balls as the origin for y is purely arbitrary. Advanced Heat and Mass Transfer. Convective Heat Transfer. Interest in studying the phenomena of convective heat and mass transfer between an ambient fluid and a body which is immersed in it stems both from fundamental considerations, such as the development of better insights into the nature of the underlying physical processes which take place, and from practical considerations,.
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