*
ﻣﻘﺪﻣﻪ اي ﺑﺮ رﺳﺎﻧﺶ Conduction: transfer of thermal energy from the more energetic particles of a medium to the adjacent less energetic ones Unlike temperature, heat transfer has direction as well as magnitude, and thus it is a vector quantity 1/86
ﻣﻘﺪﻣﻪ اي ﺑﺮ رﺳﺎﻧﺶ راﺳﺘﺎي ﺑﺮدار اﻧﺘﻘﺎل ﺣﺮارت 2/86
رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮي از زﻣﺎن و ﻣﮑﺎن ﻫﺪاﯾﺖ ﭘﺎﯾﺪار و ﮔﺬرا واﺑﺴﺘﮕﯽ ﯾﺎ اﺳﺘﻘﻼل ﻧﺮخ اﻧﺘﻘﺎل ﺣﺮارت از زﻣﺎن Heat transfer problems are often classified as being steady (also called steady-state) or transient (also called unsteady) For example, heat transfer through the walls of a house will be steady when the conditions inside the house and the outdoors remain constant for several hours. 3/86
رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮي از زﻣﺎن و ﻣﮑﺎن 4/86
رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮي از زﻣﺎن و ﻣﮑﺎن The cooling of an apple in a refrigerator, on the other hand, is a transient heat transfer process since the temperature at any fixed point within the apple will change with time during cooling temperature varies with time as well as position 5/86
رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮي از زﻣﺎن و ﻣﮑﺎن واﺑﺴﺘﮕﯽ ﯾﺎ اﺳﺘﻘﻼل ﻧﺮخ اﻧﺘﻘﺎل ﺣﺮارت ﺑﻪ ﻣﮑﺎن Lumped System: In case of variation with time but not with position, the temperature of the medium changes uniformly with time. Example: A small metal object such as a thermocouple junction or a thin copper wire Real Problems: Steady-State? Transient? Lumped? 6/86
اﻧﺘﻘﺎل ﺣﺮارت ﭼﻨﺪ ﺑﻌﺪي در دﺳﺘﮕﺎه ﻫﺎي ﻣﺨﺘﻠﻒ ﻣﺨﺘﺼﺎت ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻫﻨﺪﺳﻪ و ﻣﺎﻫﯿﺖ ﻓﯿﺰﯾﮑﯽ ﻣﺴﺄﻟﻪ 7/86
اﻧﺘﻘﺎل ﺣﺮارت ﭼﻨﺪ ﺑﻌﺪي 8/86
اﻧﺘﻘﺎل ﺣﺮارت ﭼﻨﺪ ﺑﻌﺪي Solar-panel simulation 9/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Fourier s law of heat conduction for one-dimensional heat conduction 10/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Fourier s law of heat conduction for three-dimensional heat conduction 11/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Most engineering materials are isotropic in nature, and thus they have the same properties in all directions. For such materials we do not need to be concerned about the variation of properties with direction. 12/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ But in Anisotropic materials such as the fibrous or composite materials, the properties may change with direction. In such cases the thermal conductivity may need to be expressed as a tensor quantity to account for the variation with direction. Here, we will assume the thermal conductivity of a material to be independent of direction. Anisotropic material Orthotropic material 13/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Heat Generation A medium through which heat is conducted may involve the conversion of electrical, nuclear, or chemical energy into heat (or thermal) energy. In heat conduction analysis, such conversion processes are characterized as heat generation. Thermal Simulation for CPU Cooling 14/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ heat generation is a volumetric phenomenon. That is, it occurs throughout the body of a medium. Therefore, the rate of heat generation in a medium is usually specified per unit volume. 15/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ONE-DIMENSIONAL HEAT CONDUCTION EQUATION ﻣﺨﺘﺼﺎت ﮐﺎرﺗﺰﯾﻦ 16/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Consider a thin element: thickness x, density of the wall is, specific heat is C, area of the wall normal to the direction of heat transfer is A Energy balance on this thin element during a small time interval t: 17/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 18/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﻌﺎدﻻت ﮐﻠﯽ اﻧﺘﻘﺎل ﺣﺮارت ﯾﮏ ﺑﻌﺪي در ﻣﺨﺘﺼﺎت ﮐﺎرﺗﺰﯾﻦ : thermal diffusivity = Special cases: 19/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ partial and ordinary derivatives of a function are identical when the function depends on a single variable only [T = T(x) in this case] : ﻣﺨﺘﺼﺎت اﺳﺘﻮاﻧﻪ اي 20/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Energy balance on this cylindrical shell element during a small time interval t: 21/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ or 22/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﻌﺎدﻻت ﮐﻠﯽ اﻧﺘﻘﺎل ﺣﺮارت ﯾﮏ ﺑﻌﺪي در ﻣﺨﺘﺼﺎت اﺳﺘﻮاﻧﻪ اي Special cases: 23/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﺨﺘﺼﺎت ﮐﺮوي : 24/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﻌﺎدﻻت ﮐﻠﯽ اﻧﺘﻘﺎل ﺣﺮارت ﯾﮏ ﺑﻌﺪي در ﻣﺨﺘﺼﺎت ﮐﺮوي Special cases 25/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﺷﮑﻞ ﻋﻤﻮﻣﯽ ﻣﻌﺎدﻟﻪ اﻧﺘﻘﺎل ﺣﺮارت ﯾﮏ ﺑﻌﺪي در ﻣﺨﺘﺼﺎت ﮐﺎرﺗﺰﯾﻦ اﺳﺘﻮاﻧﻪ اي و ﮐﺮوي n=0 for a plane wall n=1 for a cylinder n=2 for a sphere 26/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 27/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ASSUMTIONS Resistance wire can be considered to be a very long cylinder since its length is more than 100 times its diameter. Heat is generated uniformly in the wire and the conditions on the outer surface of the wire are uniform. Therefore, it is reasonable to expect the temperature in the wire to vary in the radial r direction only and thus the heat transfer to be 1D. Then we will have T=T(r) during steady operation. 28/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 29/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ GENERAL HEAT CONDUCTION EQUATION Rectangular Coordinates 30/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Energy balance on this element during a small time interval t: 31/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 32/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Constant thermal conductivity: Fourier-Biot Equation 33/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Cylindrical Coordinates 34/86
ﻣﻌﺎدﻻت ﺣﺎﮐﻢ ﺑﺮ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Spherical Coordinates 35/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 36/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Boundary & Initial Conditions heat flux and the temperature distribution in a medium depend on the conditions at the surfaces The mathematical expressions of the thermal conditions at the boundaries are called the boundary conditions (BCs) Solving of PDE (heat transfer equation) arbitrary constants: BCs Unique Solution 37/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 38/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Initial Conditions (ICs) Condition at the beginning of the heat conduction process = :,,,,,?,, medium is initially at a uniform temperature of Ti:,, = steady conditions: heat conduction equation does not involve any time derivatives, and thus we do not need to specify an initial condition. 39/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Boundary Conditions (BCs) Specified Temperature Specified Heat Flux Convection Radiation 40/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Specified Temperature Boundary Condition The temperature of an exposed surface can usually be measured directly and easily. The specified temperatures can be constant, which is the case for steady heat conduction, or may vary with time 41/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Specified Heat Flux Boundary Condition When there is sufficient information about energy interactions at a surface, it may be possible to determine the rate of heat transfer and thus the heat flux (heat transfer rate per unit surface area, W/m2) on that surface. The heat flux can be expressed by Fourier s law: 42/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 43/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Special Case: Insulated Boundary Some surfaces are commonly insulated in practice in order to minimize heat loss (or heat gain) through them. a well-insulated surface can be modeled as a surface with a specified heat flux of zero: temperature function must be perpendicular to an insulated surface since the slope of temperature at the surface must be zero 44/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Special Case: Thermal symmetry For example, the two surfaces of a large hot plate of thickness L suspended vertically in air will be subjected to the same thermal conditions, and thus the temperature distribution in one half of the plate will be the same as that in the other half 45/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت heat transfer problem in this plate will possess thermal symmetry about the center plane at x=l/2 no heat flow across the center plane. the center plane can be viewed as an insulated surface insulation or zero heat flux boundary condition 46/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت cylindrical (or spherical) bodies thermal symmetry about the center line (or midpoint) first derivative of temperature with respect to r (the radial variable) be zero at the centerline (or the midpoint) 47/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 48/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 49/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 50/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Convection Boundary Condition most common boundary condition Surface Energy Balance 51/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 52/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 1D heat transfer in the x-dir in a plate of thickness L T(0,t) & T(L,t)?? Note that a surface has zero thickness and thus no mass, and it cannot store any energy. Therefore, the entire net heat entering the surface from one side must leave the surface from the other side. heat continues to flow from a body to the surrounding medium at the same rate, and it just changes vehicles at the surface from conduction to convection (or vice versa) 53/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 54/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت 55/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Radiation Boundary Condition 56/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت In some cases, a heat transfer surface is surrounded by an evacuated space and thus there is no convection heat transfer between a surface and the surrounding medium. In such cases, radiation becomes the only mechanism of heat transfer between the surface under consideration and the surroundings. 1D heat transfer in the x-dir in a plate of thickness L 57/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Interface Boundary Condition layers of different materials : ﺗﻌﯿﯿﻦ ﺷﺮاﯾﻂ ﻣﺮزي در ﻧﺎﺣﯿﻪ در ﺗﻤﺎس دو ﻣﺤﯿﻂ ﺑﻪ دو ﻓﺮض زﯾﺮ واﺑﺴﺘﻪ اﺳﺖ 1. same temperature at the area of contact 2. an interface (which is a surface) cannot store any energy, and thus the heat flux on the two sides of an interface must be the same 58/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Perfect Contact 59/86
ﺷﺮاﯾﻂ ﻣﺮزي و اوﻟﯿﻪ ﺣﺎﮐﻢ ﺑﺮ ﻣﻌﺎدﻻت Imperfect Contact results in thermal contact resistance, which is considered in the next chapter 60/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Solution of steady 1D heat conduction problems ﻣﺴﺄﻟﻪ اﻧﺘﻘﺎل ﺣﺮارت ﻣﻌﺎدﻟﻪ اﻧﺘﻘﺎل ﺣﺮارت ﺣﺎﮐﻢ ﺑﺮ ﻣﺴﺄﻟﻪ ﺑﺎ در ﻧﻈﺮ ﮔﺮﻓﺘﻦ ﺗﺮم ﻫﺎي ﻣﺨﺘﻠﻒ ﺣﺮارﺗﯽ )ﺑﺪﺳﺖ آوردن ﻣﻌﺎدﻟﻪ دﯾﻔﺮاﻧﺴﯿﻞ ﺣﺎﮐﻢ( ﺣﻞ ﻋﻤﻮﻣﯽ ﻣﻌﺎدﻟﻪ دﯾﻔﺮاﻧﺴﯿﻞ اﻋﻤﺎل ﺷﺮاﯾﻂ ﻣﺮزي 61/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 62/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻓﺮﺿﯿﺎت : رﺳﺎﻧﺎﯾﯽ ﯾﮏ ﺑﻌﺪي رﺳﺎﻧﺎﯾﯽ ﭘﺎﯾﺎ ﺛﺎﺑﺖ k no heat gen. 63/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 64/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 65/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 66/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 67/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻓﺮﺿﯿﺎت : رﺳﺎﻧﺎﯾﯽ ﯾﮏ ﺑﻌﺪي رﺳﺎﻧﺎﯾﯽ ﭘﺎﯾﺎ ﺛﺎﺑﺖ k no heat gen. no radiation 68/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 69/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 70/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 71/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 72/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﻌﺎدﻟﻪ ﺗﻮزﯾﻊ دﻣﺎ در ﯾﮏ اﺳﺘﻮاﻧﻪ 73/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ = ln ln + = ( ) ( ) 74/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﻌﺎدﻟﻪ ﺗﻮزﯾﻊ دﻣﺎ در ﯾﮏ ﮐﺮه 75/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ = ( ) 76/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 77/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Composite Media 78/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ 79/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Wire: Twire(r=0)=? But TI,wire=Twire(r=r1)?? T1? Heat transfer equation on ceramic Ceramic: 80/86
ﻣﺴﺎﺋﻞ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ Interface B.C. 81/86
ﺿﺮﯾﺐ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮ When the variation of thermal conductivity with temperature in a specified temperature interval is large, however, it may be necessary to account for this variation to minimize the error. 82/86
ﺿﺮﯾﺐ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮ 83/86
ﺿﺮﯾﺐ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮ 84/86
ﺿﺮﯾﺐ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮ 85/86
ﺿﺮﯾﺐ اﻧﺘﻘﺎل ﺣﺮارت رﺳﺎﻧﺎﯾﯽ ﻣﺘﻐﯿﺮ 86/86