Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 10
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The equations are first made dimensionless. For demonstration purposes, let us consider the relatively simple problem of a binarymixture with constant properties and density flowing at low speed, where body forces, heatsource term, and chemical reactions are neglected. The conservation equations are, from Eqs.1.42, 1.46, 1.58, and 1.63,MassV •V = 0(1.64)DVp~- - V P + IaV2VMomentumDTThermal energypc ~DmlDtSpecies= kV2T + ~t~_(1.65)(1.66)(1.67)DV2m 1Using L and V as characteristic length and velocity, respectively, we define the dimensionlessvariablesx*-V* -XLvVy* - yLz*- zL(1.68)(1.69)1.24CHAPTERONEtt* -P* -and also(1.70)L/VP(1.71)pV 2T-T~T* = ~(1.72)T=-T~(1.73)m * = m l - ml,wml** - ml.wwhere the subscript oo refers to the external free-stream condition or some average conditionand the subscript w refers to conditions adjacent to a bounding surface across which transferof heat and mass occurs. If we introduce the dimensionless quantities (Eqs.
1.68-1.73), intoEqs. 1.64-1.67, we obtain, respectively,V* • V* = 0(1.74)DV*-V'P*Dt* -D T*Dt*1Re PrDm*1Dt*Re Sc- - ~Jr"1(1.75)V*2V *V * 2 T * -.[.-2 EcReO*V*2m*(1.76)(1.77)Obviously, the solutions of Eqs. 1.74-1.77 depend on the coefficients that appear in theseequations. Solutions of Eqs. 1.74-1.77 are equally applicable to the model and prototype(where the model and prototype are geometrically similar systems of different linear dimensions in streams of different velocities, temperatures, and concentration), if the coefficients inthese equations are the same for both model and prototype. These coefficients, Pr, Re, Sc, andEc (called dimensionless parameters or similarity parameters), are defined in Table 1.10.Focusing attention now on heat transfer, from Eq.
1.14, using the dimensionless quantities,the heat transfer coefficient is given ask 3T*]h-L3y* y,=o(1.78)or, in dimensionless form,h L _ 3T*]k= Nu(1.79)~y* r=0where the dimensionless group Nu is known as the Nusselt number. Since Nu is the dimensionless temperature gradient at the surface, according to Eq. 1.76 it must therefore dependon the dimensionless groups that appear in this equation; henceNu = fl(Re, Pr, Ec)(1.80)For processes in which viscous dissipation and compressibility are negligible, which is the casein many industrial applications, we haveNu = f2(Re, Pr)(forced convection)(1.81)0t-I0• ,,,.ir~00II, Id"u•I=I"~n~..•-~=~:•~:•~...I.~~1~~ ~010.~~~,~~,~.-~~.,-~~~~.,• ,..~'~og...c:-~-~¢~°~o=~°~ooI~o=~~o~o~oo-~~°ol=lo~ol:l~~~~~"-.0•°~1~~10,o~,.~~~~~c~~:~~0~:~~~,~o.-~1o~~0,~00,~0~1-0°'~~...~~~,~0--~~.,,,~~,.~..o",~..o~"~~..oO~o"~..o~~..o~~~.~oo.,.~~~~~oI~ooo°~'II1~11--:o ~"~IIv• ,~~°~¢~~~JrO~o~1~ ~~~•o,-~~ ~,~.,..~01.251.26%r~0.,=,~00Ill==10~000~'~•~ ~o~~.~~ ~.~0.,=~oII0.,.,~00o0>000o00~~00~~.~0~.~ ~~o0.,..~o0o¢)o0~'~~~.,.,~II0~.o.~~sIIo0o0ooo3"0II0°,.,~2~II0a3a3oo0ooor~or~r~a30o0:~°.,~o~a30~00a3l-qr~~8~0.,..~000.£1-i0,.~~0=000.
,.,,~IIz~0oIIr~=CC000o0or~0o.,..~00,.- , ~~"~,E,.c:oown~"~ow~J.,.q wI,~Eo,.clo~8*~•' - ~'~..~.1.271.28CHAPTERONEIn the case of buoyancy-induced flow, Eq. 1.65 should be replaced with the simplified version[16] of Eq. 1.50, and, following a similar procedure, we should obtainNu = f3(Gr, Pr)(natural convection)(1.82)where Gr is the Grashof number, defined in Table 1.10. Also, using the relation of Eq. 1.17and dimensionless quantities,h a - D /)m* II(1.83)L /)Y* ly*:oL - Om*hD D3y* Iy*:0 = Shor(1.84)This parameter, termed the Sherwood number, is equal to the dimensionless mass fraction(i.e., concentration) gradient at the surface, and it provides a measure of the convection masstransfer occurring at the surface. Following the same argument as before (but now for Eq.1.77), we haveSh = fa(Re, Sc)(forced convection, mass transfer)(1.85)The significance of expressions such as Eqs.
1.80-1.82 and 1.85 should be appreciated. Forexample, Eq. 1.81 states that convection heat transfer results, whether obtained theoreticallyor experimentally, can be represented in terms of three dimensionless groups instead of sevenparameters (h, L, V, k, Cp, It, and p). The convenience is evident. Once the form of the functional dependence of Eq.
1.81 is obtained for a particular surface geometry (e.g., from laboratory experiments on a small model), it is known to be universally applicable, i.e., it may beapplied to different fluids, velocities, temperatures, and length scales, as long as the assumptions associated with the original equations are satisfied (e.g., negligible viscous dissipationand body forces). Note that the relations of Eqs. 1.80 and 1.85 are derived without actuallysolving the system of Eqs.
1.64-1.67. References 3, 7, 12, 15, 16, and 18 cover the above procedure in more detail and also include many different cases.It is important to mention here that once the conservation equations are put in dimensionless form it is also convenient to make an order-of-magnitude assessment of all terms inthe equations. Often a problem can be simplified by discovering that a term that would bevery difficult to handle if large is in fact negligibly small [7, 12]. Even if the primary thrust ofthe investigation is experimental, making the equations dimensionless and estimating theorders of magnitude of the terms is good practice.
It is usually not possible for an experimental test to include (simulate) all conditions exactly; a good engineer will focus on the mostimportant conditions. The same applies to performing an order-of-magnitude analysis. Forexample, for boundary-layer flows, allowance is made for the fact that lengths transverse tothe main flow scale with a much shorter length than those measured in the direction of mainflow. References 7, 12, and 17 cover many examples of the order-of-magnitude analysis.When the governing equations of a problem are unknown, an alternative approach ofderiving dimensionless groups is based on use of dimensional analysis in the form of theBuckingham pi theorem [9, 11, 14, 16, 18].
The success of this method depends on our abilityto select, largely from intuition, the parameters that influence the problem. For example,knowing in advance that the heat transfer coefficient in fully developed forced convection ina tube is a function of certain variables, that is, h = f(V, p, kt, Cp, k, D), we can use the Buckingham pi theorem to obtain Eq. 1.81, as shown in Ref.
11. However, this method is carriedout without any consideration of the physical nature of the process in question, i.e., there is noway to ensure that all essential variables have been included. However, as shown above, starting with the differential form of the conservation equations we have derived the similarityparameters (dimensionless groups) in rigorous fashion.BASIC CONCEPTS OF HEAT TRANSFER1.29In Table 1.10 those dimensionless groups that appear frequently in the heat and masstransfer literature have been listed.
The list includes groups already mentioned above as wellas those found in special fields of heat transfer. Note that, although similar in form, the Nusselt and Biot numbers differ in both definition and interpretation. The Nusselt number isdefined in terms of thermal conductivity of the fluid; the Biot number is based on the solidthermal conductivity.UNITS AND CONVERSION FACTORSThe dimensions that are used consistently in the field of heat transfer are length, mass, force,energy, temperature, and time.
We should avoid using both force and mass dimensions in thesame equation, since force is always expressible in dimensions of mass, length, and time, andvice versa. We do not make a practice of eliminating energy in terms of force times length,because the accounting of work and heat is practically always kept separate in heat transferproblems.In this handbook both SI (the accepted abbreviation for Systdme International d'Unit~s, orInternational System of Units) and English engineering units* are used simultaneouslythroughout.
The base units for the English engineering units are given in the second columnof Table 1.11. The unit of force in English units is the pound force (lbf). However, the use ofthe pound mass (Ibm) and pound force in engineering work causes considerable confusion inthe proper use of these two fundamentally different units.TABLE 1.11QuantityC o n v e r s i o n Factor gc for the C o m m o n Unit SystemsSIEnglish engineering*cgs*Metric engineeringMasskilogram, kgp o u n d mass, Ibmgram, gkilogram mass, kgLengthTimeForcemeter, msecond, snewton, N1 kg.m/(N.s2) ~foot, ftsecond, s, or hour, hpound force, lbf32.174lbm.ft/(lbf.s2)centimeter, cmsecond, sdyne, dyn1 g.crn/(dyn.s2)meter, msecond, skilogram force, kgf9.80665kg.rn/(kgf.s2)gc* In this system of units the temperature is given in degrees Fahrenheit (°F).* Centimeter-gram-second: this system of units has been used mostly in scientific work.* Since 1 kg.m/s 2 = 1 N, then gc = 1 in the SI system of units.The two can be related as1 Ibm × 32.174 ft/s 21 lbf =whencegcgc = 32.174 lbm-ft/(lbf.s 2)Thus, gc is merely a conversion factor and it should not be confused with the gravitationalacceleration g.
The numerical value of gc is a constant depending only on the system of unitsinvolved and not on the value of the gravitational acceleration at a particular location. Values* Also associated with this system of units are such names as U.S. Customary Units, British engineering units, engineering units, and foot-pound-second system of units. The name English engineering units, or, for short, English units, isselected in this handbook because it has been used by practicing engineers more frequently than the other names mentioned.1.30CHAPTER ONETABLE 1.12SI Base and Supplementary UnitsQuantityUnitLengthMassTimeElectric currentThermodynamic temperatureAmount of substanceLuminous intensityPlane angle*Solid angle*meter (m)kilogram (kg)second (s)ampere (A)kelvin (K)mole (mol)candela (cd)radian (rad)steradian (sr)* Supplementary units.of gc corresponding to different systems of units found in engineering literature are given inTable 1.11.The SI base units are summarized in Table 1.12.
The SI units comprise a rigorouslycoherent form of the metric system, i.e., all remaining units may be derived from the baseunits using formulas that do not involve any numerical factors. For example, the unitof force is the newton (N); a 1-N force will accelerate a 1-kg mass at 1 m/s 2. Hence 1 N =1 kg.m/s 2. The unit of pressure is the N/m 2, often referred to as the pascal. In the SI systemthere is one unit of energy (thermal, mechanical, or electrical), the joule (J); 1 J = 1 N.m.The unit for energy rate, or power, is joules per second (J/s), where one J/s is equivalent toone watt (1 J/s = 1 W).In the English system of units it is necessary to relate thermal and mechanical energy viathe mechanical equivalent of heat J~ ThusJc x thermal energy = mechanical energyThe unit of heat in the English system is the British thermal unit (Btu).
When the unit ofmechanical energy is the pound-force-foot (lbcft), thenJc = 778.16 lbf-ft/Btuas I Btu - 778.16 lbf.ft. Happily, in the SI system the units of heat and work are identical andJc is unity.Since it is frequently necessary to work with extremely large or small numbers, a set ofstandard prefixes has been introduced to simplify matters (Table 1.13). Symbols and namesfor all units used in the handbook are given in Table 1.14.
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