Mathematical model related to heat distribution 5

Building a model is the key to applying the formula.

1. Introduction With the wide application of various mulching materials in agriculture (in 1988, the coverage area of plastic film in China reached more than 200 x 10'ha, and the coverage area of sand field and orange straw exceeded x 10'ha"') The study of the mulching benefit, that is, the study of the impact of mulching on soil hydrothermal status, fertility, salinity, weed growth, diseases and pests, etc. Among them, the study of soil hydrothermal transport is the basis for studying other effects. In the past 20 years, a large number of studies have been carried out on the effects of various mulching materials in various countries around the world, most of which are experimental studies on the effects of soil temperature. Due to the different climates and geographical conditions, it is difficult to promote the scientific research results in one place and one place to other countries and regions; And the experimental cycle is long, which consumes a lot of manpower and material resources.

In recent years, some researchers have introduced mathematical modeling methods into this field, and gratifying progress has been made. In 1979, Mahrer et al. first conducted a one-dimensional mathematical simulation study on the temperature distribution of soil under transparent plastic film, and then simulated the coupled water-heat transport and two-dimensional heat migration of soil under transparent film, and concluded that the warming value gradually increased with the increase of soil water content under plastic film covering2 and the temperature gradient at the covering edge was large. This paper lays a solid theoretical foundation for the analysis of the warming and water retention effect of plastic films.

Chen Fazu*], Institute of Geography, Chinese Academy of Sciences, is also hypothetically ignoring the heat exchange between the cover and the surface of the air layer and the heat conduction between the upper and lower surfaces of the cover. This article is 6 pages in total) [Continue reading this article].

The heat transfer model, which is a partial differential equation, can be understood as a diffusion process.

There is a book called "Partial Differential Equations", which mainly deals with the construction and solution of similar partial differential equations.

The heat transfer model is a partial differential equation that can be understood as a diffusion process.

A book called "Partial Differential Equations" is mainly involved in the construction methods and solutions of similar partial differential equations.

The heat transfer from a fixed point to a fixed point is heat conduction, the heat transfer at the flow point is heat convection, and the heat transfer in the form of waves is heat radiation.

1. Heat conduction, mainly expressed by Fourier's law.

Heat transfer rate Heat transfer area = – thermal conductivity.

Temperature gradient. 2. Thermal convection, mainly expressed by Newton's cooling law.

Convective heat transfer rate Heat transfer area = - convective heat transfer coefficient * temperature difference between the wall surface and the main body of the fluid.

3. Thermal radiation, mainly in the form of blackbody radiation.

Equation (Stephan-Boltzmann equation).

Blackbody emission capability = radiation to the 4th power of Changshu * absolute temperature.

1. Definition: Heat conduction is a heat transfer phenomenon when there is no macroscopic movement in the medium, which can occur in solids, liquids and gases, but strictly speaking, only in solids is pure heat conduction, and even if the fluid is at rest, it will also produce natural convection due to the density difference caused by the temperature gradient, therefore, heat convection and heat conduction occur at the same time in the fluid.

Thermal convection, also known as convective heat transfer, refers to the heat transfer process caused by the relative displacement of particles in the fluid, and is one of the three ways of heat transfer.

Thermal radiation, the phenomenon in which an object radiates electromagnetic waves due to having a temperature. One of the 3 ways in which heat is transferred. All objects with temperatures above absolute zero can produce thermal radiation, and the higher the temperature, the greater the total energy radiated and the more shortwave components will be emitted.

2. Difference: Heat conduction is the process of transferring heat energy from high temperature to low temperature; Thermal convection is the process by which heat is transferred through a flowing medium; Thermal radiation is the phenomenon in which an object radiates electromagnetic waves due to its temperature and is the only way to transfer heat in a vacuum.

The solution of the thermal equation has the characteristic of smoothing the initial temperature.

Self-priming, which means that the hot bai travels from high temperature to low temperature. In general, the initial state of many different DAOs will tend to the same steady state (thermal equilibrium). Therefore, it is difficult to deduce the initial state from the existing heat distribution, even for very short time intervals.

The thermal equation is also the simplest example of a parabolic partial differential equation.

Using the Laplace operator, the heat equation can be generalized to the following form.

where δ is the Laplace operator for spatial variables.

The heat equation governs heat conduction and other diffusion processes, such as particle diffusion or nerve cell action potentials. The heat equation can also be used as a model for certain financial phenomena, such as the Black-Scholes model and the Ornstein-Uhlenbeck process. The thermal equation and its nonlinear generalization are also used in image analysis.

Although the Schrödinger equation in quantum mechanics has a mathematical formula similar to the thermal equation (but the time parameter is a pure imaginary number), the essence is not a diffusion problem, and the qualitative behavior of the solution is completely different.

Technically, the thermal equation defies the special theory of relativity because its solution expresses that a perturbation can propagate across space in an instant. The effect of perturbations outside the front light cone is usually negligible, but to derive a reasonable velocity for heat conduction, a hyperbolic partial differential equation must be considered instead.

A: When pancakes are baked in a square pan, the heat is concentrated in the corners and the food is scorched right in the corners (and even the edges). In a round pan the heat will be evenly distributed throughout the outer edges, and the food will not be scorched by the edges.

However, because most ovens are rectangular, it is not as efficient to use a round pan. Build a model to represent the distribution of heat on the outer edges of different shapes of pans – from rectangular to circular and in the middle.

Try building a model that shows the distribution of heat through the outer edge of different pot bases: square to round and other shapes in between.

Assumptions: 1The square oven has a width-to-length ratio of W L;

2.All reference pots must have an area of a;

3.Initially, the two stands of the oven are placed equally.

Build a model to filter the best pot type in the following scenarios:

1.the maximum number of pots suitable for the oven (n);

2.Pot type to maximize uniform heat distribution (h);

3.The optimal conditions (1) and (2), with the respective occupancy ratios of p and (1-p), are used to describe the difference between w l and p.

In addition to providing a standard MCM format solution, a 1-2 page advertising campaign for Brownie Food Magazine will require you to highlight your design and results.

b: Scarcity of available freshwater resources.

The scarcity of freshwater resources has become a bottleneck for the development of many countries around the world.

Establish a mathematical model of a country's 2013 water strategy, identify an efficient, realistic, and cost-effective water strategy to meet the country's (United States, China, Russia, Egypt or Arabia, whichever chooses) the projected water needs by 2025, and determine the optimal water strategy. In particular, the mathematical model you build must take into account the country's water reserves and flows, the development of seawater and freshwater treatment, and the state of water conservation. If possible, apply the model you have established to discuss the economic, geographical, and environmental implications of the model, and provide the country's leadership with a non-technical position paper that outlines your approach, its feasibility and costing, and why it is the "best strategic choice."

Selectable countries: United States, China, Russia, Egypt or Saudi Arabia.

2013 Mathematical Modeling Competition.

MCM issues.

Question A: The Ultimate Brownie Pan.

When condensed in the 4 corners of a rectangular pan when baking hot, and around the corners (and to a lesser extent at the edges): the product will overdo it. The heat in a circular disc is evenly distributed throughout the outer rim and at the edges of the product is not overheaded.

However, because most ovens use round pans, the shape is rectangular and is not very efficient relative to the space used in the oven.

Develop a model to show the heat distribution across the outer edges of the different shapes of the pan – between the rectangles – and the other shapes.

Hypothesis 1. The ratio of width to length of w l is a rectangular oven.

2。Each plate must have an area of A.

3。Initially, two racks in the oven, evenly spaced.

Build a model that can be used to select the best pan type (shape) in the following cases:

1。Fit in the oven in a pot that can maximize the number (n).

2。Maximize the uniform distribution of heat (h), pan.

3。Optimize the combination of conditions (1) and (2) in the weights p and (for 1 - p) are assigned to the result to illustrate how with different values of w l and p.

In addition to the MCM format solution, prepare a one- to two-page advertising film for the new Brownie Food Magazine highlighting its design and results.

Question B: Water, water, everywhere.

Fresh boiled water is constrained by the development constraints in most parts of the world. To build a mathematical model for identifying effective, feasible and cost-effective water resources strategies by 2013 to meet projected water needs, select a country from the list below to determine the optimal water strategy by 2025. In particular, your mathematical model must address storage and movement, desalinization, and protection.

If possible, use your model to influence economic, physical, and environmental impacts on your strategy. Provide a non-technical position paper that leads to an introduction to your approach, its feasibility and cost, and why it is the "best water strategy choice." ”

The countries are: USA, China, Russia, Egypt, Saudi Arabia.

You don't have to ask your mentor specifically.

Urban green space extraction and land surface temperature inversion do not knock **, it can also be done in software such as ENVI or ERDAS, of course, it is best to understand IDL programming, because IDL batch processing is still very useful when the data volume is particularly large.

There is only one topic, others must not be clear, it is best to communicate with your mentor, it may be that your mentor asked you to make a system, which needs to be developed, so you need to be familiar with software engineering.

In this study, the traditional mathematical model of groundwater flow with mixed boreholes is no longer applicable, and the following mathematical model (Chen Chong-xi, Jiao J J, 1999 et al.) is used to describe the three-dimensional unstable flow model of groundwater with mixed boreholes in the Shule River Basin

Investigation and evaluation of the rational development and utilization of groundwater resources in the Shule River Basin of the Hexi Corridor.

where: h is the head function of the aquifer or weakly permeable layer, m; h0 is the initial head function of the study area, m; h1

is the known head function of the first type boundary in the study area, m; zsp is the elevation of the spring mouth, m; μs

is the unit water storage coefficient of the aquifer or weakly permeable layer, 1 m; d is the gravity feed degree of the unpressurized aquifer; qw

vw is the production volume of the mining well and the volume of the wellbore working section; w is the algebraic sum of the infiltration recharge intensity of atmospheric rainfall and infiltration, such as rivers, canal systems, field irrigation, etc., and the evaporation intensity of the submerged surface, m d; q is the flow function per unit area of the second type of boundary in the study area, m d; bl

It is the first type boundary of the study area; b2

It is the second type of boundary of the study area; d is the distribution range of the study area; khkz is the horizontal and vertical permeability coefficient of the aquifer or weakly permeable layer, m d; kze is the vertical equivalent permeability coefficient, m d.

When there is a linear flow in the mixing wellbore:

Investigation and evaluation of the rational development and utilization of groundwater resources in the Shule River Basin of the Hexi Corridor.

where: d is the inner diameter of the filter tube, m; for the gravity of the fluid; is the dynamic viscosity of the fluid;

When there is a nonlinear flow in the mixed wellbore:

Investigation and evaluation of the rational development and utilization of groundwater resources in the Shule River Basin of the Hexi Corridor.

where: f is the coefficient of friction; v is the osmotic flow rate.

Incidentally, the formulation of the boundary conditions of the diving surface in the above mathematical model is different from that of Bear J (1983) in most literatures, due to the latter error.