Determining heat transfer coefficients. Dimensionless numbers

4 Determining heat transfer coefficients. Dimensionless numbers 

Knowledge of the temperature field in the fluid is a prerequisite for the calculation of the heat transfer coefficient using (1.25). This, in turn, can only be determined when the velocity field is known. Only in relatively simple cases, exact values for the heat transfer coefficient can be found by solving the fundamental partial differential equations for the temperature and velocity. Examples of this include heat transfer in fully developed, laminar flow in tubes and parallel flow over a flat plate with a laminar boundary layer. Simplified models are required for turbulent 

An important method for finding heat transfer coefficients was and still is the experiment. By measuring the heat flow or flux, as well as the wall and fluid temperatures the local or mean heat transfer coefficient can be found using (1.25) and (1.33). To completely solve the heat transfer problem all the quantities which influence the heat transfer must be varied when these measurements are taken. These quantities include the geometric dimensions (e.g. tube length and diameter), the characteristic flow velocity and the properties of the fluid, namely density, viscosity, thermal conductivity and specific heat capacity. 

The number of these variables is generally between five and ten. To quantify the effect of one particular property, experiments must be done with at least n (e.g. n = 5) different values whilst keeping all other variables constant. With m different variables to consider in all, the number of individual experiments required will be nm. With six variables and n = 5 then 56 = 15625 experimental runs will have to be done. Obviously this demands a great deal of time and expense. 

Velocity and temperature fields are therefore only similar when also the dimensionless groups or numbers concur. These numbers contain geometric quantities, the decisive temperature differences and velocities and also the properties of the heat transfer fluid. The number of dimensionless quantities is notably smaller than the total number of all the relevant physical quantities. The number of experiments is significantly reduced because only the functional relationship between the dimensionless numbers needs to be investigated. Primarily, the values of the dimensionless numbers are varied rather than the individual quantities which make up the dimensionless numbers. 

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The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

Effectively, Eqs. (86) and (87) describe two interpenetrating continua which are thermally coupled. The value of the heat transfer coefficient a depends on the specific shape of the channels considered suitable correlations have been determined for circular or for rectangular channels [100]. In general, the temperature fields obtained from Eqs. (86) and (87) for the solid and the fluid phases are different, in contrast to the assumptions made in most other models for heat transfer in porous media [117]. Kim et al. [118] have used a model similar to that described here to compute the temperature distribution in a micro channel heat sink. They considered various values of the channel width (expressed in dimensionless form as the Darcy number) and various ratios of the solid and fluid thermal conductivity and determined the regimes where major deviations of the fluid temperature from the solid temperature are found. 

The final dimensionless group to be evaluated is the interfacial heat-transfer number, and therefore the interfacial heat-transfer coefficient and the interfacial area must be determined. The interface is easily described for this regime, and, with a knowledge of the holdup and the tube geometry, the interfacial area can be calculated. The interfacial heat trasfer coefficient is not readily evaluated, since experimental values for U are not available. A conservative estimate for U is found by treating the interface as a stationary wall and calculating U from the relationship... 

For practical purposes, heat-transfer engineers often use empirical or semi-empirical correlations to predict h values. These formulations are usually based on the dimensionless numbers described before. In this case, the appropriate formulation should be used to prevent significant errors. If dimensionless correlations are applicable under conditions of gas extraction, then heat-transfer coefficients can be determined from these correlations and the influence of parameter variations may be derived also from them. 

The determination of heat transfer coefficients with the assistance of dimensionless numbers has already been explained in section 1.1.4. This method can also be used for mass transfer, and as an example we will take the mean Nusselt number Num = amL/ in forced flow, which can be represented by an expression of the form... 

Finally, it should also be pointed out that in heat conduction problems the dimensionless representation and the combination of the influencing quantities into dimensionless numbers are not as significant as in the representation and determination of heat transfer coefficients in 1.1.4. In the following sections we will frequently refrain from making the heat conduction problem dimensionless and will only present the solution of a problem in a dimensionless form by a suitable combination of variables and influencing quantities. 

Rohsenow et al. [4.19] rearranged (4.51) by introducing dimensionless quantities and then determined the heat transfer coefficient a = AL/<5 and also the mean heat transfer coefficient am from the calculated film thickness. Fig. 4.14 shows as a result of this the mean Nusselt number... 

To determine the process side heat transfer coefficient, hg, empirical correlations for many impeller t3q)es and tank geometries have been established. The parameterizations are generally expressed in terms of dimensionless numbers, geometrical ratios and viscosity ratios (see e.g., [23, 65, 40] and references therein). 

The dimensionless parameter determining the limiting mode of heat transfer is the Biot number Bi, defined in terms of k, L and surEace heat transfer coefficient h ... 

Buller and Kilburn (1981) performed experiments determining the heat transfer coefficients for laminar flow forced air cooling for integrated circuit packages mounted on printed wiring boards (thus for conditions differing from that of a flat plate), and correlated he with the air speed through use of the Colburn J factor, a dimensionless number, in the form of... 

The stirring and the resulting flow pattern inside the tank can be very important for the overall heat transfer resistance, because the performance of the reactor affects the heat transfer coefficient at the process side ho- The other resistances are determined by the materials used and the properties of the cooling/heating media and are thus not influenced by the reactor performance. To determine the process side heat transfer coefficient, ho, empirical correlations for many impeller types and tank geometries have been established. The parameterizations are generally expressed in terms of dimensionless numbers, geometrical ratios and viscosity ratios (see e.g., [23, 40, 65] and references therein). 

Maintenance of proper temperature is a major aspect of reactor operation. The illustrations of several reactors in this chapter depict a number of provisions for heat transfer. The magnitude of required heat transfer is determined by heat and material balances as described in Section 17.3. The data needed are thermal conductivities and coefficients of heat transfer. Some of the factors influencing these quantities are associated in the usual groups for heat transfer namely, the Nusselt, Stanton, Prandtl, and Reynolds dimensionless groups. Other characteristics of particular kinds of reactors also are brought into correlations. A selection of practical results from the abundant literature will be assembled here. Some modes of heat transfer to stirred and fixed bed reactors are represented in Figures 17.33 and 17.18, and temperature profiles in... 

The major problem is determination of heat and mass transfer coefficients. Similitude theory allows the definition Of two dimensionless coefficients including h. and h. These are the so called Nusselt and Stanton numbers given by ... 

In conclusion, it is worthwhile to cite two additional dimensionless parameters describing process of heat and mass transfer at convective flow of a medium. Determine first the heat-transfer coefEdent a as ratio of thermal flux to the temperature difference, that is q = oAT, and then the mass-transfer coefficient as the ratio of the diffusion flux to difference of concentrations j = AC. Then one can introduce two dimensionless Nusselt s numbers... 

Another important consideration in cooling channel design is to ensure that the coolant circulates in turbulent rather than laminar (streamline) flow. The coefficient of heat transfer of the cooling system is drastically reduced in laminar flow. The condition of laminar or turbulent flow is determined by the Reynolds number (Re). This is a dimensionless number given by the equation ... 

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