Heat flux expressions

Conductive heat transfer does not require any motion of atoms or molecules, as only the interactions between atoms or molecules transfers heat. The heat flux expressed in W rn 2 can be described using Fourier s law ... 

C Consider transient onc-dimensional heat conduction in a plane wall that is to be solved by the explicit method. If both sides of the wall ate subjected to specified heat flux, express the stability criterion for this problem in its simplest form. 

Due to axisymetric geometry, there are no lateral fluxes. In the reference case, the contact between the canister and the clay is supposed to be perfect, which implies non-existence of gas and water flow between them. To model the canister heating, a time dependent heat flux, expressed in W, is imposed at that boundary. It follows equation 1 ... 

For the natural convection case, in general, the rate of diffusion of the water vapor will not be great enough to sufficiently overcome the loss of frost through mechanical failure and removal of condensed portions in the boundary layer, so the qmin point will not be reached [ ]. Therefore, the heat flux expressions and correlations in this paper cannot be used for this case unless the frost surface temperature can be determined. 

Species mass concentration Tensor in heat-flux vector expression Species contribution to extra stress tensor Potential energy for all molecules in liquid Tensor used in heat-flux expression Potential energy for single molecule Potential energy for single molecule in external field... 

While the conveetive heat flux expression is straightforward, the expression for drying rate needs explanation. The drying rate ean be ealculated from this formula, when drying is eontrolled by gas-side resistance. The driving force is then the difference between absolute humidity at equilibrium with solid surface and that of bulk gas. When solid surface is saturated with moisture, the expression for Y is identical to Equation 3.48 when solid surface contains bound moisture, Y will result from Equation 3.46 and a sorption isotherm. This is in essence the so-called equilibrium method of drying rate calculation. 

The first term of the above heat flux expression q f, t) is equal to zero as the integral over all directions is equal to zero due to the local thermal equilibrium. Using equation (8.59), we can write... 

The lack of correlation between the flucUiating stress tensor and the flucUiating heat flux in the third expression is an example of the Curie principle for the fluctuations. These equations for flucUiating hydrodynamics are arrived at by a procedure very similar to that exliibited in the preceding section for difllisioii. A crucial ingredient is the equation for entropy production in a fluid... 

Heat transfer by nucleate boiling is an important mechanism in the vaporization of liqmds. It occurs in the vaporization of liquids in kettle-type and natural-circulation reboilers commonly usea in the process industries. High rates of heat transfer per unit of area (heat flux) are obtained as a result of bubble formation at the liquid-solid interface rather than from mechanical devices external to the heat exchanger. There are available several expressions from which reasonable values of the film coefficients may be obtained. 

Using the expression for A appearing in equation 9.62, we find that, to first-order, the heat-flux Q is given by... 

This is derived by substituting from the heat-balance expression, Eq. (20). Now, the skewed-cosine heat-flux profile being considered gives a known functional relationship between the flux and the quality at any position along the channel. By equating this relationship with Eq. (35), a solution can be obtained giving the local values of (j> and k at the predicted burn-out position. The corresponding peak flux can then be evaluated, and in this way the predicted burn-out lines for the three mass velocities in Fig. 40 can be drawn. 

Here, J (a, t) is the instantaneous interfacial flux expressed as a state vector whose components are the diffusional and heat fluxes. Therefore, the expected value < J(a)> takes into account the variations in residence time among the entire bubble population. 

The value of the integral in the energy balance (equation 11.55) is again given by equation 11.60 [substituting (6S - 8o) for 0 ]. The heat flux q0 at the surface is now constant, and the right-hand side of equation 11.55 may be expressed as (—qa/Cf,p). Thus, for constant surface heat flux, equation 11.55 becomes ... 

What is the equivalent expression for the instantaneous heat flux under analogous conditions ... 

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

In the buffer zone the value of d +/dy+ is twice this value. Obtain an expression for the eddy kinematic viscosity E in terms of the kinematic viscosity (pt/p) and y+. On the assumption that the eddy thermal diffusivity Eh and the eddy kinematic viscosity E are equal, calculate the value of the temperature gradient in a liquid flowing over the surface at y =15 (which lies within the buffer layer) for a surface heat flux of 1000 W/m The liquid has a Prandtl number of 7 and a thermal conductivity of 0.62 W/m K. 

The conditions on the interface express the continuity of the mass and heat fluxes and the equilibrium of all acting forces (Landau and Lifshitz 1959). In the frame of reference associated with the interface they are ... 

When the temperature Ts of the interface is constant, and wall heat flux is also constant, temperature oscillations are the result of the meniscus displacement along micro-channel axis. They are expressed as... 

There are two causes for oscillations of the heat flux, with 7 = const. (1) fluctuations of the heat transfer coefficient due to velocity fluctuations, and (2) fluctuations of the fluid temperature. At small enough Reynolds numbers the heat transfer coefficient is constant (Bejan 1993), whereas at moderate Re (Re 10 ) it is a weak function of velocity (Peng and Peterson 1995 Incropera 1999 Sobhan and Garimella 2001). Bearing this in mind, it is possible to neglect the influence of velocity fluctuations on the heat transfer coefficient and assume that heat flux flucmations are expressed as follows ... 

The constant of proportionality k is known as the thermal conductivity of the material and the above relationship is known as Fourier s law for conduction in one dimension. The thermal conductivity k is the heat flux which results from unit temperature gradient in unit distance. In s.i. units the thermal conductivity, k, is expressed in Wm"1 K. Integration of Fourier s law yields... 

The total heat flux from the entire boiling surface can be expressed as... 

For alkali metals with small cavities at low pressures, the value of / for a given heat flux may not be achievable. Since tw can be expressed as a function of average heat transfer rate per unit area, <7", and liquid properties, Eq. (2-112) can be rearranged and solve for the heat flux ... 

In inverted annular flow dryout, liquid mass flux is low enough and wall heat fluxes are high enough to cause vapor to be generated rapidly near the wall, forming a vapor annulus surrounding a liquid core (Fig. 4.176). The vapor generation near the wall occurs so quickly that the velocities of the two phases are about equal, or S = 1, so the expression for the void fraction at dryout, ado, can be calculated from the known dryout quality, Ydo ... 

Because the components of the analytical expression for C are not sufficiently known to permit an analytical evaluation, C is determined empirically as a function of the local quality at the point of DNB, XDNB, (under nonuniform heat flux conditions) and the bulk mass flux, G. The empirically determined expression for C is... 

Bowring CHF correlation for uniform heat flux (Bowring, 1972). For water in round tubes with uniform heat flux, the CHF can be expressed as... 

One long side of the compartment wall was split into a large number of thin, horizontal strips and the heat flux from the gas layer to the center of each strip calculated using the well known expression... 

The emission coefficient was taken to be a constant value close to unity. The configuration factor, F, was calculated in a conventional way, treating the center of each strip as a point. Once the downward flame spread started the radiation from the wall flames and the pyrolysing lining material behind the flames was added to the smoke layer radiation. The heat flux to the walls was then calculated from the expression... 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

---