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Heat Flow
Published in Samuel C. Sugarman, HVAC Fundamentals, 2020
Heat is energy in the form of molecules in motion. As a substance becomes warmer, its molecular motion and energy level (temperature) increases. Temperature describes the level of heat (energy) with reference to no heat. Heat is a positive value relative to no heat. Because all heat is a positive value in relation to no heat, cold is not a true value. It is really an expression of comparison. Cold has no number value and is used by most people as a basis of comparison only. Therefore, warm and hot are comparative terms used to describe higher temperature levels. Cool and cold are comparative terms used to describe lower temperature levels. The Fahrenheit scale is the standard system of temperature measurement used in the United States. The U.S. is one of the few countries in the world still using this system. Most countries use the metric temperature measurement system, which is the Celsius scale. The Fahrenheit and Celsius scales are currently used interchangeably in the U.S. to describe equipment and fundamentals in the heating, ventilating and air conditioning industry.
Physics
Published in Keith L. Richards, Design Engineer's Sourcebook, 2017
While the Celsius and Fahrenheit scales are widely used today, there have been other scales developed have included the Rankine scale, the Newton scale, and the Romer scale. All of these are rarely used today. A further scale, the Kelvin temperature scale, has been adopted as the standard metric system of temperature measurement and is possibly the widest-used scale by engineers and scientists. The Kelvin scale is similar to the Celsius scale in that there are 100 divisions between the freezing and boiling points of water. However, the zero-degree mark on the Kelvin scale is 273.15 units cooler than it is on the Celsius scale. Hence, the temperature of 0 K is equivalent to a temperature of −273.15°C. Note that the degree symbol is not used on the Kelvin scale, so that a temperature of 250 units above 0 K is referred to as 250 K and not as 250° 0 K; the temperature is abbreviated as 250 K. Conversions between Celsius temperatures and Kelvin temperatures can be performed using one of the two following equations:
Essentials of Data Analytics
Published in Adedeji B. Badiru, Data Analytics, 2020
The standardized structure and decimal features of the metric system made it well suited for scientific and engineering work. Consequently, it is not surprising that the rapid spread of the system coincided with an age of rapid technological development. In the United States, by the Act of Congress in 1866, it became “lawful throughout the United States of America to employ the weights and measures of the metric system in all contracts, dealings or court proceedings.” However, the United States has remained a hold-out with respect to a widespread adoption of the metric system. Today, in some localities of the United States, both English and metric systems are used side by side.
Modified Tseng's extragradient methods for variational inequality on Hadamard manifolds
Published in Applicable Analysis, 2021
Junfeng Chen, Sanyang Liu, Xiaokai Chang
Let ∇ be the Levi-Civita connection associated with the Riemannian metric. Let γ be a smooth curve in . A vector field X along γ is said to be parallel if , where θ is the zero tangent vector. If itself is parallel along γ, we say that γ is a geodesic (this notion is different from the corresponding one in the calculus of variations), and in this case is constant. When , γ is said to be normalized. A geodesic joining x to y in is said to be minimal if its length equals . A Riemannian manifold equipped with Riemannian distance d is a metric space .
Energy decay rate for the wave equation with variable coefficients and boundary source term
Published in Applicable Analysis, 2021
Set For each , we define the inner product and the norm on the tangent space by Then is a Riemannian manifold with Riemann metric g. and are denoted by the gradient of u and Levi-Civita connection in the Riemannian metric g, respectively. It follows that Let H be a vector field on . Then the covariant differential of H determines a bilinear form on , for each , by where is the covariant derivative of the vector field H with respect to X.
Weak sharpness and finite termination for variational inequalities on Hadamard manifolds
Published in Optimization, 2021
Let and be the Riemannian manifold with the Riemannian metric The Riemannian distance is given by The sectional curvature of M is zero and it holds that M is a Hadamard manifold. For each , the tangent plane at x equals to . The unique geodesic γ starting from with velocity is defined by Thus, Moreover, for any , we have Hence, the inverse of exponential map is defined as