The CRC Handbook of Mechanical Engineering. Chapter 4. Heat and Mass Transfer (776127), страница 11
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With heating or cooling of the fluid, there may or may not be a change in thephase of the fluid. Here, only heat transfer to or from a single-phase fluid is considered.Fully Developed Velocity and Temperature Profiles. When a fluid enters a tube from a large reservoir,the velocity profile at the entrance is almost uniform as shown in Figure 4.2.21. The fluid in the immediatevicinity of the tube surface is decelerated and the velocity increases from zero at the surface to uc at adistance d from the surface; in the region r = 0 to (R – d) the velocity is uniform.
The value of d increasesin the direction of flow and with constant fluid density the value of the uniform velocity uc increases.At some location downstream, d reaches its maximum possible value, equal to the radius of the tube,and from that point onward the velocity profile does not change.The region where d increases, i.e., where the velocity profile changes, is known as the entrance regionor hydrodynamically developing region. The region downstream from the axial location where d reachesits maximum value and where the velocity profile does not change is the fully developed velocity profileor hydrodynamically fully developed region.
Similarly, downstream of the location where heating orcooling of the fluid starts, the temperature profile changes in the direction of flow. But beyond a certaindistance the dimensionless temperature profile does not change in the direction of flow. The region wherethe dimensionless temperature profile changes is the thermally developing region or the thermal entranceregion, and the region where the dimensionless temperature profile does not change is the thermally© 1999 by CRC Press LLC4-47Heat and Mass TransferFIGURE 4.2.21 Developing and fully developed velocity profiles.fully developed region. For simultaneously developing velocity and temperature profiles in laminar flows,the hydrodynamic and thermal entrance lengths are given byLe= 0.0565Re ddLe,thdLe,thd= 0.053Re d Pr= 0.037RePr(4.2.88)Uniform heat flux(4.2.89)Uniform surface temperature(4.2.90)In most engineering applications, with turbulent flows, correlations for fully developed conditions canbe used after about 10 diameters from where the heating starts.Convective Heat Transfer Coefficient and Bulk Temperature.
The reference temperature for defining theconvective heat transfer coefficient is the bulk temperature Tb and the convective heat flux is given byq ¢¢ = h(Ts - Tb )(4.2.91)The bulk temperature Tb is determined from the relationTbrvC TdAò=ò rvC dAAcAcppc(4.2.92)cwhere Ac is the cross-sectional area perpendicular to the axis of the tube.If the fluid is drained from the tube at a particular axial location and mixed, the temperature of themixed fluid is the bulk temperature.
It is also know as the mixing cup temperature. With heating orcooling of the fluid the bulk temperature varies in the direction of flow. In some cases we use the termmean fluid temperature, Tm, to represent the arithmetic mean of the fluid bulk temperatures at inlet andexit of the tube.© 1999 by CRC Press LLC4-48Section 4Heat Transfer CorrelationsLaminar Flows — Entrance Region. For laminar flows in a tube with uniform surface temperature, inthe entrance region the correlation of Sieder and Tate (1936) is13æ Re Pr ö æ m öNu d = 1.86ç d ÷ ç ÷è L d ø è ms ø0.14(4.2.93)valid forL Re d Pr æ m ö<d8 çè m s ÷ø0.420.48 < Pr < 16, 7000.0044 <m< 9.75msThe overbar in the Nusselt number indicates that it is formed with the average heat transfer coefficientover the entire length of the tube.
Properties of the fluid are evaluated at the arithmetic mean of the inletand exit bulk temperatures. In Equation (4.2.93) the heat transfer coefficient was determined fromT + Tbe öæq = h pdLç Ts - bi÷è2 ø(4.2.94)Therefore, to find the total heat transfer rate with h from Equation (4.2.93) employ Equation (4.2.94).Laminar Flows — Fully Developed Velocity and Temperature Profiles. Evaluate properties at the bulktemperatureUniform Surface TemperatureNu d = 3.66( 4.2.95)Uniform Surface Heat FluxNu d = 4.36( 4.2.96)Turbulent Flows. If the flow is turbulent, the difference between the correlations with uniform surfacetemperature and uniform surface heat flux is not significant and the correlations can be used for bothcases. For turbulent flows, Gnielinsky (1976, 1990) recommends:Evaluate properties at the bulk temperature.0.6 < Pr < 20002300 < Re d < 10 6Nu d =0 < d L <1( f 2) (Re d - 1000) Pr é æ d ö 2 3 ùê1 +ú121 + 12.7( f 2) (Pr 2 3 - 1) ë è L ø û[]f = 1.58 ln(Re d ) - 3.28-2(4.2.97)(4.2.98)f = friction factor = 2tw/rv2.To reflect the effect of variation of fluid properties with temperature, multiply the Nusselt numbersin Equation (4.2.97) by (Tb/Ts)0.45 for gases and (Pr/Prs)0.11 for liquids where the temperatures are absolute,and T and Pr with a subscript s are to be evaluated at the surface temperature.
The equations can beused to evaluate the heat transfer coefficient in the developing profile region. To determine the heat© 1999 by CRC Press LLC4-49Heat and Mass Transfertransfer coefficient in the fully developed region set d/L = 0. A simpler correlation (fully developedregion) is the Dittus–Boelter (1930) equation. Evaluate properties at Tb.0.7 £ Pr £ 160Re d > 10, 000d L > 10Nu d = 0.023Re 4d 5 Pr n(4.2.99)where n = 0.4 for heating (Ts > Tb) and n = 0.3 for cooling (Ts < Tb).For liquid metals with Pr ! 1 the correlations due to Sleicher and Rouse (1976) areUniform surface temperature:Nu d ,b = 4.8 + 0.0156 Re 0d.,85f Prs0.93(4.2.100)Nu d ,b = 6.3 + 0.0167Re 0d.,85f Prs0.93(4.2.101)Uniform heat flux:Subscripts b, f, and s indicate that the variables are to be evaluated at the bulk temperature, filmtemperature (arithmetic mean of the bulk and surface temperatures), and surface temperature, respectively.In the computations of the Nusselt number the properties (evaluated at the bulk temperature) vary inthe direction of flow and hence give different values of h at different locations.
In many cases arepresentative average value of the convective heat transfer coefficient is needed. Such an average valuecan be obtained either by taking the arithmetic average of the convective heat transfer coefficientsevaluated at the inlet and exit bulk temperatures or the convective heat transfer coefficient evaluated atthe arithmetic mean of the inlet and exit bulk temperatures. If the variation of the convective heat transfercoefficient is large, it may be appropriate to divide the tube into shorter lengths with smaller variationin the bulk temperatures and evaluating the average heat transfer coefficient in each section.Uniform Surface Temperature — Relation between the Convective Heat Transfer Coefficient and theTotal Heat Transfer Rate: With a uniform surface temperature, employing an average value of theconvective heat transfer coefficient the local convective heat flux varies in the direction of flow.
To relatethe convective heat transfer coefficient to the temperatures and the surface area, we have, for the elementallength Dz (Figure 4.2.22).ṁC pdTbdA= h s (Ts - Tb )dzdzFIGURE 4.2.22 Elemental length of a tube for determining heat transfer rate.© 1999 by CRC Press LLC(4.2.102)4-50Section 4Assuming a suitable average convective heat transfer coefficient over the entire length of the tube,separating the variables, and integrating the equation from z = 0 to z = L, we obtainlnTs - TbehAs=˙ pTs - TbimC(4.2.103)Equation (4.2.103) gives the exit temperature. For a constant-density fluid or an ideal gas, the heattransfer rate is determined from˙ p (Tbe - Tbi )q = mC(4.2.104)Equation (4.2.103) was derived on the basis of uniform convective heat transfer coefficient.
However,if the functional relationship between h and Tb is known, Equation (4.2.102) can be integrated bysubstituting the relationship. The convective heat transfer coefficient variation with Tb for water in twotubes of different diameters for two different flow rates is shown in Figure 4.2.23.
From the figure it isclear that h can be very well approximated as a linear function of T. By substituting such a linear functionrelationship into Equation (4.2.102), it can be shown thatlnhAhi Ts - Tbe=- s s˙ phe Ts - TbimC(4.2.105)where hi, he, and hs are the values of the convective heat transfer coefficient evaluated at bulk temperaturesof Tbi, Tbe, and Ts, respectively. Although it has been demonstrated that h varies approximately linearlywith the bulk temperature with water as the fluid, the variation of h with air and oil as the fluid is muchsmaller and is very well approximated by a linear relationship.
For other fluids it is suggested that therelationship be verified before employing Equation (4.2.105). [Note: It is tempting to determine the heattransfer rate from the relationq = hAs(Ts- Tbe ) + (Ts - Tbi )2Replacing q by Equation (4.2.104) and solving for Tbe for defined values of the mass flow rate and tube˙ p > 2. Use of Equationsurface area, the second law of thermodynamics will be violated if hAs / mC(4.2.103) or (4.2.105) ensures that no violation of the second law occurs however large As is.]Uniform Surface Heat Flux: If the imposed heat flux is known, the total heat transfer rate for adefined length of the tube is also known. From Equation (4.2.104) the exit temperature of the fluid isdetermined.
The fluid temperature at any location in the pipe is known from the heat transfer rate up tothat location (q = q²As) and Equation (4.2.104). The convective heat transfer coefficient is used to findthe surface temperature of the tube.Temperature Variation of the Fluid with Uniform Surface Temperature and Uniform Heat Flux: Thefluid temperature variations in the two cases are different. With the assumption of uniform heat transfercoefficient, with a uniform surface temperature the heat flux decreases in the direction of flow leadingto a progressively decreasing rate of temperature change in the fluid with axial distance.
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