Form Of Continuity Equation Research Articles

Performance of a natural ventilation system with heat recovery in UK classrooms: An experimental study

This paper presents the ventilation performance of a passive ventilation system with heat recovery (PVHR) based on in-situ monitoring in a primary school in London. The study involves long-term (15-month) monitoring of temperature, relative humidity and carbon dioxide (CO2) concentrations in both the classrooms and the outdoor environment. In addition, short term (1- and 2-week) observational monitoring was performed in two classrooms at ventilation system level and classroom level, during both the heating and non-heating seasons. Temperatures and air velocities were measured within the PVHR system while instances of window opening and the number of students were noted in daily diaries. Air permeability and infiltration measurements were performed to characterize the spaces. Time-varying ventilation rates were estimated through a form of continuity equation considering CO2 generation rates by occupants. Preliminary results show that the operation of the ventilation system is more sensitive to changes in wind speed and direction than to buoyancy. When negative pressure was observed on the classrooms’ facades the ventilation system was supplying two to three times more air in comparison to instances when positive pressures were observed. The assessment of the ventilation performance of such natural ventilation systems depending solely on wind and buoyancy is complicated as they are dynamic systems that constantly balancing with the surrounding conditions, and the operation is highly correlated to the airtightness of the building's envelope.

Verification of a Chemical Nonequilibrium Flows Solver Using the Method of Manufactured Solutions

This paper presents code verification of a chemically nonequilibrium flows solver using the method of manufacturedsolutions.The Method of Manufactured Solutions(MMS) is a general approach for creating exact solutions to the governing equations and can be used in the code verification process. In the MMS, the analytical solutions for the flow variables are first constructed, then the governing equations are modified to satisfy these solutions by adding appropriate source terms which are generated by applying the governing equations to these solutions. After that, code verification process will start. The order of accuracy of the calculations will be computed and compared with theoretical order of accuracy to determine if the code passes the verification test. We created manufactured solutions for two different sets of Euler equations. One set includes the total density equation plus ns-1 species equations and the other contains only species continuity equations. The results show that the form of continuity equations has little influence on the behaviour of global conservative variable errors as the mesh is refined. Our study also indicates that a complete chemical reaction model is preferred to ensure the convergence of observed order of accuracy is smooth in the order of accuracy test.

Numerical simulation of liquid absorption in paper‐like swelling porous media

AbstractThe work is built on a previous research by Wiryana and Berg, in which wicking into four wet‐formed paper stripes, consisting of cellulose fibers and four different percentages of the powdered carboxymethyl cellulose (CMC) superabsorbent, was studied experimentally. Because of the swelling of cellulose fibers and CMC powder on contact with water, the wicking was accompanied by a swelling of the matrix. A finite element/control volume (FE/CV)‐based computer program is used for the first time to model the wicking in such swelling porous medium. The simulation used a novel form of continuity equation, which included the effects of liquid absorption and matrix swelling, in conjunction with the Darcy's law to model the single‐phase flow behind a clearly defined liquid‐front. A new method of estimating the time‐varying permeability of the paper, based on the absorbed liquid‐mass vs. time plots, is also proposed. Later, this time‐dependent permeability is used in the numerical simulation to change the permeability in elements behind the moving liquid‐front as a function of the time that the element has been wetted by the liquid, since the passage of the liquid‐front. The numerical prediction of the wicking‐front location as a function of time compares well with the reported experimental data. © 2011 American Institute of Chemical Engineers AIChE J, 58: 2536–2544, 2012

A Nonlinear PDE-Based Method for Sparse Deconvolution

In this paper, we introduce a new nonlinear evolution partial differential equation for sparse deconvolution problems. The proposed PDE has the form of continuity equation that arises in various research areas, e.g. fluid dynamics and optimal transportation, and thus has some interesting physical and geometric interpretations. The underlying optimization model that we consider is the standard $\ell_1$ minimization with linear equality constraints, i.e. $\min_u\{\|u\|_1 : Au=f\}$ with $A$ being an under-sampled convolution operator. We show that our PDE preserves the $\ell_1$ norm while lowering the residual $\|Au-f\|_2$. More importantly the solution of the PDE becomes sparser asymptotically, which is illustrated numerically. Therefore, it can be treated as a natural and helpful plug-in to some algorithms for $\ell_1$ minimization problems, e.g. Bregman iterative methods introduced for sparse reconstruction problems in [W. Yin, S. Osher, D. Goldfarb, and J. Darbon,SIAM J. Imaging Sci., 1 (2008), pp. 143-168]. Numerical experiments show great improvements in terms of both convergence speed and reconstruction quality.

Conservation laws and Hamilton’s equations for systems with long-range interaction and memory

Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action principle: generalized Noether’s theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time–space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional differential action 1-form and find the corresponding Hamiltonian type equations from the closed condition of the form.

Linear and angular coherence momenta in the classical second-order coherence theory of vector electromagnetic fields

A new concept of vector and tensor densities is introduced into the general coherence theory of vector electromagnetic fields that is based on energy and energy-flow coherence tensors. Related coherence conservation laws are presented in the form of continuity equations that provide new insights into the propagation of second-order correlation tensors associated with stationary random classical electromagnetic fields.

Continuity equations, H-theorems, and maximum entropy

Linear partial differential evolution equations that have the form of continuity equations are investigated with reference to the temporal behavior of the Gibbs-Shannon-Jaynes entropy functional S( t) = − ∫ f (ln f) d x associated with their solutions (regarded as probability distributions f( x, t)). Many attempts have been made to approximately solve these type of equations, based upon Jaynes' maximum entropy principle. We show that, within this maximum entropy approach, the functional relation d S[ f ME]/d t = F [ f ME(x, t)], giving the time derivative of the entropy in terms of the approximate maximum entropy solution, has exactly the same form as the corresponding relation d S[ f exact]/d t = F [ f exact(x, t)] that hold in the case of the exact solutions. From this it follows that any H-theorem (involving the Gibbs-Shannon-Jaynes entropy) satisfied by the exact solutions of the differential equations holds for the maximum entropy approximations f ME( x, t) as well. Some illustrative examples are discussed.

Continuity equations for many-body systems

The four-divergence of a generalized current is related to the Green’s function formulation of the many-body problem. The relations obtained take the form of continuity equations, with an extra source term corresponding to the creation or annihilation of particles allowed by the second-quantization formalism. With the aid of these many-body continuity equations, a connection between the two types of conservation laws for Green’s function in the literature, the generalized Ward identity and the conserving approximations of Baym and Kadanoff is derived.