Energy equation boundary layer, laminar

This is the energy equation of the laminar boundary layer. The left side represents the net transport of energy into the control volume, and the right side... 

What is the energy equation for the laminar boundary layer on a flat plate What assumptions are involved in the derivation of this equation ... 

The temperature of a liquid metal stream discharged from the delivery tube prior to primary breakup can be calculated by integrating the energy equation in time. The cooling rate can be estimated from a cylinder cooling relation for the liquid jet-ligament breakup mechanism (with free-fall atomizers), or from a laminar flat plate boundary layer relation for the liquid film-sheet breakup mechanism (with close-coupled atomizers). 

Take into consideration two-dimensional, rectilinear, steady, incompressible, constant-property, laminar boundary layer flow in the x direction along a flat plate. Assume that viscous energy dissipation may be neglected. Write the continuity, momentum and energy equations. 

If terms of the order (8/L)2 and less are again neglected, it will be seen that the energy equation for laminar two-dimensional boundary layer flow becomes ... 

The energy integral equation is applied in basically the same way as the momentum integral equation. The form of the boundary layer temperature profile, i.e., of the variation of (T - T ) with y, is assumed. In the case of laminar flow, for example, a polynomial form is again often used. The unknown coefficients in this assumed temperature profile are then determined by applying known boundary conditions on temperature at the inner and outer edges of the boundary layer. For example, the variation of the wall temperature Tw with x may be specified. Therefore, because at the outer edge of the boundary layer the temperature must become equal to the freestream temperature T, two boundary conditions on the assumed temperature profile in this case are ... 

If terms of the order (SIL) and less are again neglected and itf it is assumed that the Prandtl number, Pr, has order 1 or greater, i.e., is not small,j it follows that the x-wise diffusion term, i.e., d2T/dx2. is negligible compared to the other terms. Hence, the energy equation for free convective laminar two-dimensional boundary layer flow becomes ... 

The foregoing analysis considered the fluid dynamics of a laminar-boundary-layer flow system. We shall now develop the energy equation for this system and then proceed to an integral method of solution. 

We have already seen that the phenomenological laws governing heat, mass, and momentum transfer are similar. In Chap. 5 it was shown that the energy and momentum equations of a laminar boundary layer are similar, viz.. 

We wish to obtain a solution to the laminar-boundary-layer momentum and energy equations, assuming constant fluid properties and zero pressure gradient. We have ... 

Next we apply three fundamental laws to this fluid element Conservation of mass, conservation of momentum, and conservation of energy to obtain the continuity, momentum, and energy equations for laminar flow in boundary layers. 

In this section we derive the equation of motion that governs the natural convection flow in laminar boundary layer. The conservation of mass and energy equations derived in Chapter 6 for forced convection are also applicable for natural convection, but tlie momentum equation needs to be modified to incorporate buoyancy. 

In laminar flow, heat transfer occurs only by conduction, as there are no eddies to carry heat by convection across an isothermal surface. The problem is amenable to mathematical analysis based on the partial differential equations for continuity, momentum, and energy. Such treatments are beyond the scope of this book and are given in standard treatises on heat transfer, Mathematical solutions depend on the boundary conditions established to define the conditions of fluid flow and heat transfer. When the fluid approaches the heating surface, it may have an already completed hydrodynamic boundary layer or a partially developed one. Or the fluid may approach the heating surface at a uniform velocity, and both boimdary layers may be initiated at the same time. A simple flow situation where the velocity is assumed constant in all cross sections and tube lengths is called... 

Similar solutions for Prandtl numbers other than unity may be obtained from Eqs. 6.117 and 6.118 or their equivalent. A major simplification is the independence of the momentum equation (Eq. 6.117), from the energy equation (Eq. 6.118), which makes/independent of /. Also, the linear form of the energy equation in / permits handling arbitrary surface temperature distributions as in the case of the flat plate. (See the section on the two-dimensional laminar boundary layer.)... 

Nonuniform Surface Temperature. Nonuniform surface temperatures affect the convective heat transfer in a turbulent boundary layer similarly as in a laminar case except that the turbulent boundary layer responds in shorter downstream distances The heat transfer to surfaces with arbitrary temperature variations is obtained by superposition of solutions for convective heating to a uniform-temperature surface preceded by a surface at the recovery temperature of the fluid (Eq. 6.65). For the superposition to be valid, it is necessary that the energy equation be linear in T or i, which imposes restrictions on the types of fluid property variations that are permitted. In the turbulent boundary layer, it is generally required that the fluid properties remain constant however, under the assumption that boundary layer velocity distributions are expressible in terms of the local stream function rather than y for ideal gases, the energy equation is also linear in T [%]. 

Laminar Free Convection. Sparrow and Gregg [33] were the first to use the boundary layer method to study laminar, gravity-driven film condensation on a vertical plate. They improved upon the approximate analysis of Nusselt by including fluid acceleration and energy convection terms in the momentum and energy equations, respectively. Their numerical results can be expressed as ... 

Formulate the equations of continuity, momentum, energy, and continuity of component A for the corresponding boundary layers over a flat plate in laminar flow. 

If T terface and Tbuik replace Ca, equilibrium and Ca, bulks respectively, in the definition of the dimensionless profile P, and the thermal diffusiv-ity replaces a. mix. then the preceding equation represents the thermal energy balance from which temperature profiles can be obtained. The tangential velocity component within the mass transfer boundary layer is calculated from the potential flow solution for vg if the interface is characterized by zero shear and the Reynolds number is in the laminar flow regime. Since the concentration and thermal boundary layers are thin for large values of the Schmidt and Prandtl... 

First let us examine mass transport through this film under isothermal conditions by employing the continuity equations for mass (a mass balance) and for momentum (an energy balance). In this stagnant film, which can correspond to the laminar boundary layer that develops when a fluid passes over a flat surface, there is no motion of the fluid, hence the latter equation is irrelevant. The continuity equation for mass describes the spacial dependence of concentration in terms of the velocities parallel, u, and perpendicular, V, to the surface ... 

The continuity equations for mass, x-direction momentum, chemical species and energy in the plane, stationary, laminar boundary layer flow have already been given as Eqs. (7.1) to (7.4). The stream function ij/, by means of which the mass continuity equation is automatically satisfied, is defined by Eqs. (7.5). Following the approaches of Lees (1956), Fay and Riddell (1958), and Chung (1965), self-similar solutions in the stagnation region are obtained via transformations from (x, y) co-ordinates to the two new variables... 

---