Linear temperature profile

Parameter values for seeded batch cooling crystallizer 

Parameters Value from experiments / model fitting 

This system has been studied by Hu et al.[99], Shi et al.[100], and Yenkie et al.[98], and the data for this system is given below. 

---

Change the model and program to account for a linear temperature profile. 

For a storage fluid which is thermally stratified with a linear temperature profile in the vertical direction, the energy content can be shown with Equation (5) to be... 

The coefficient of thermal conductivity can be defined in reference to the experiment shown schematically in Fig. 12.2. In this example the lower wall (at z = 0) is held at a fixed temperature T and the upper wall (at z = a) is held at some higher temperature T + AT. At steady state there will be a linear temperature profile across the gap, with temperature gradient dT/dz = AT/a. Heat will flow from the hot wall toward the colder wall, and the heat flux q is proportional to the areas of the plates, proportional to the temperature... 

Again, neglecting viscous heating results in the linear temperature profile, eqn. (6.249), presented in the previous section. In this case, with the imposed pressure gradient, the equation of motion becomes... 

Both shear thinning and temperature dependence of viscosity strongly affect the melting rate. Their effect on the rate of melting can be estimated by considering a case in which convection is neglected and viscous dissipation is low enough to permit the assumption that the viscosity variation across the film is determined by a linear temperature profile ... 

Eq. (10) can be approximately integrated through analytical techniques similar to Eq. (7) over the time increment Atn either by assuming a constant average particle temperature between Tj(tn) and Tj(tn i) or by using a linear temperature profile in this range. 

As a first example of low-density heat transfer let us consider the two parallel infinite plates shown in Fig. 12-14. The plates are maintained at different temperatures and separated by a gaseous medium. Let us neglect natural-convection effects. If the gas density is sufficiently high so that A — 0, a linear temperature profile through the gas will be experienced as shown for the case of A. As the gas density is lowered, the larger mean free paths require a greater distance from the heat-transfer surfaces in order for the gas to accommodate to the surface temperatures. The anticipated temperature profiles are shown in... 

Alternatively, the duration of hydrothermal convection in the oceanic crust can be estimated by mapping the distribution of nonlinear temperature profiles taken during heat-flow measurements as a function of oceanic-crustal age. Purely conductive heat loss (i.e., no hydrothermal circulation) results in linear temperature profiles in sediments, while convective heat loss results in concave or convex profiles, depending on whether the water penetrates into or comes out of the sediments. 

Fig. 1.5 shows (x) and the deviation A (x) from the linear temperature profile between wi and W2- At high temperatures, where the thermal conductivity is large, the temperature gradient is smaller than at lower temperatures, where A( ) is smaller. At each point... 

The formation of the condensate flow dM, requires a heat flow dQ - Ahv dM to be removed, where Ahv is the enthalpy of vaporization. This heat will, by presumption, be transferred purely by heat conduction through the condensate film. The heat transferred through the condensate film by convection can be neglected. If, in addition to this, we presume constant thermal conductivity of the condensate and with that a linear temperature profile, like that in Fig. 4.6, then along surface segment b dx the heat flow will be... 

Figure 43 Piece-wise linear temperature profile.

In terms of the assumed linear temperature profile, the heat transfer is... 

For forced convection of liquid metals (Pr 1) over a horizontal flat plate subject to a uniform heat flux qm, evaluate the Nusselt number based on a uniform velocity and linear temperature profiles (Eg. 5P-2). 

In terms of a linear temperature profile, we have already obtained the constant involved with Eq, (5.58) to he C — 0.289 (see Table 5.3). Also, the exact relation is known to be... 

Neglecting convection effects in the film (i.e., assuming pure conduction in the film, which yields a linear temperature profile), an energy balance on a differential slice of condensate of width dz (Fig. 14.4) gives... 

Rohsenow [13] showed that if the condensate temperature profile was allowed to be nonlinear to account for convection effects in the condensate film, an improved correction term, i tg = itg + 0.68c,f( Ts - TJ) results. Another correction pertains to the variation of viscosity with temperature. For the assumed linear temperature profile in the condensate, Drew [14] showed that if l/pf is linear in temperature, then the condensate viscosity should be calculated at a reference temperature equal to T, - Yt(T, - Tw). 

A second iteration is performed in a manner similar to the first iteration, but using the last estimate of product compositions and estimating the feed-zone temperature from a linear temperature profile between the distillate and bottoms. This temperature is 88.2°F (31.2°C) and results in a distillate rate of 519.7 Ibmole/hr. 

It is possible to model the deformation of film bubbles with a system of dimensionless equations that is derived according to the following assumptions [13] steady-state and axisymmetrical flow (z-axis) of an incompressible fluid thin and flat film external forces on the bubble are neglected Newtonian, pseudoplastic, or viscoelastic fluids and linear temperature profiles between die exit and freezeline position. The system of dimensionless fundamental equations can be represented, irrespective of the rheological constitutive equation used, as shown in the following equations ... 

In the case of blown film simulations, a linear temperature profile can be used to obtain a greater stability in the solution of the system. The set of boundary conditions imposed on the system is given as shown in the following equation ... 

A theoretical model for the heterogeneous nucleation was proposed by Hsu [10] for the growth of pre-existing nuclei in a cavity on a heated surface. The model included the effect of nmi-uniform superheated liquid. The equation for the activation curve of bubble nucleation was derived by combining the Clausius-Qapeyron and the Young—Laplace equations. Then, by substituting the linear temperature profile into the equation, the range of active cavity sizes on the heated surface was obtained. 

A more realistic prediction of the melting performance can be obtained if the polymer melt is considered non-Newtonian and non-isothermal. However, this extension of the analysis results in coupled energy and momentum equations. Such problems generally do not allow analytical solutions. One approach to this problem, as suggested by Tadmor [61], is to assume a certain temperature profile and solve the equations. If a quadratic temperature profile is assumed, the solution becomes quite elaborate containing many error functions. Evaluation of the solutions requires substantial numerical analysis and number crunching for details, the reader is referred to reference 5 of Chapter 1. If a linear temperature profile is assumed in the melt him, the solution becomes more manageable. The constitutive equation is ... 

The integration constant corresponds to the normal coordinate where the shear stress is zero and the velocity profile has an extremum. If it is assumed that the viscosity is determined by a linear temperature profile the velocity gradient can be written as ... 

The solution for the velocity profile and flow rate is correct as long as the viscous dissipation can be neglected. The solution for the temperature profile is approximate because of the assumption of a linear temperature profile for viscosity determination. if the calculated temperature profile deviates sharply from the linear profile the error in the temperature profile can be significant. This issue will be addressed in the following section. 

Figure 10.7 shows a simple one-dimensional model of the contact between two materials sliding over another at constant speed. Material 1, which is assumed to dissipate most of the heat exhibits a linear temperature profile. The amount of thermal... 

The model predicts peak temperatures about 0.5 metres from the top of the reactor for the liquid/gas interface, the bulk liquid and, to a lesser extent, for the gas-phase. A conversion of about 90% is attained at 1 metre from the top. The organic liquid/gas interfacial temperature was obtained by assuming a linear temperature profile in the film, which conflicts with the assumption of turbulent flow of the organic phase. 

Ruorescent Theimometiy, Rgure 5 (a) Schematic of the microfluidic device used by Natrajan and Christensen [7] Id obtain a one-dimensional, steady-state temperature gradient (b) Variation of temperature across the microtluidic device shown in (a). The solid circles represent the mean temperatures obtained via two-dye LIF measurements while the solid line indicates the predicted linear temperature profile across the microchannel anay. The span of the error bars represents twice the standard deviation for each mean temperature. Adapted from [7]... 

---