A general steady state mathematical model for fin-and-tube heat

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化工专业英语单词汇总

a function of …的函数absorption：吸附acetone 丙酮acrylics丙烯酸树脂Aerospace 航空agricultural engineering农业工程agricultural engineer农艺师Amalgam 汞齐，水银；混合物，交叉ammonia 氨ammonium nitrate硝酸铵ammonium sulfate硫酸铵analyte分析物analytical chemistry分析化学amorphous 非定型的,非晶型的,非结晶的,玻璃状的;无一定目的的,乱七八糟approximate to：接近，趋近area 面积argon氩aromatic 芳香烃的as a whole整体而言ash纯碱asphalt沥青a priori：先验的,既定的,不根据经验的,由原因推出结果的,演绎的,直觉的accessory heater 附属加热器accident prevention事故预防accountant会计师，会计，出纳activity coefficient活度系数actualrate of absorption 实际吸收速率adiabatic绝热的，不传热的alkane烷烃ammonia-air mixture 氨气-水混合物ammonium phosphate磷酸铵anhydrous无水的applied Chemistry应用化学aquatic plant 水生植物artificial人工的asphaltene沥青油assay分析化验at right angles to 与…成直角，与…垂直bottoms product塔底产品baffle-plate折流挡板，缓冲挡板balance 抵消，平衡barrier障碍物batch间歇的；benzene苯binary distillation双组分精馏bioengineering生物工程bionics（仿生学）biosynthesis生物合成blower 风机boundary layer 边界层brick wall 墙壁brittleness 脆性bubble-cap tower 泡罩塔Buchner funnel 布氏漏斗bulk explosive集装炸药buoyancy force 浮力by virtue of 由于，根据，凭借于barrel桶(国际原油计量单位)base塔底，基础biological production生物制品生产biomechanics生物力学bitumen沥青blood-flow dynamics血液流动动力学boiling point 沸点bottom 底部，塔底branched chain支链烷烃branched-chain（带支链的）bulk chemical 大宗化工产品capillary action毛细管作用carbon dioxide 二氧化碳capital expenditure 基建投资carbon skeleton碳骨架capital outlay 费用，成本，基建投资carrier载体carbon tetrachloride四氯化碳straightforward简单明了的catalyst 催化剂catalyst催化剂，catalytic cracking 催化裂化catalytic oxidation催化氧化chemical additive添加剂centrifuge离心.离心机，离心分离chemical process safety 化工过程安全chain-shaped链状的chemical reactor transfer of heat, evaporation, crystallization结晶chain链c hlorofluorocarbon二氯二氟化碳，氟里昂chemical reaction化学反应c irculating gas 循环气civil engineer土木工程师closed system封闭系统cleansing agent清洗剂compound化合物close teamwork紧密的团队协作computer microchip 计算机芯片coefficient系数concentration difference 浓度差columnar liquid chromatography柱状液相色谱仪concentration gradient 浓度梯度combustion燃烧condensate冷凝液，凝缩液commercial proportions 商业规模condensation冷凝commodity or specialty通用商品或特殊化学品condenser冷凝器compress压缩condense凝缩，冷凝computationally intensive计算量大的constitute取代物，取代基concave （凸的，凸面）continuous：连续的concentration 浓度convection 对流conduction 传导convection drying对流干燥conduit导流管）, cooling water 冷却水conical funnel 锥形漏斗coordinating ligand配合体，向心配合体conservation of mass and energy能量与质量守衡定律corrosive property 腐蚀性constant-rate drying period恒速干燥阶段cost engineer造价师control volume 控制体counterpart对应物，配对物cooler 冷却器cracking of petroleum石油裂解counteract抵消cracking裂化counter-current逆流criteria 指标conversion转化，转化率critical moisture content临界湿湿含量crude oil原油cake filtration 饼层过滤capillarity毛细现象，毛细管力crystal flaw 晶体瑕疵decompose分解，离解，还原，腐烂depth 深度derivative衍生物descriptive 描述性的diagnostic application of computers计算机诊断diammonium hydrogen phosphate磷酸二氢铵dislocation 错位dispersion model分散模型dissolve 溶解distillate product塔顶馏出产品distillate section精馏段downcomer 降液管drag forces曳力drying of Solids 固体干燥drying rate干燥速率ductility 延展性dairy牛奶deactivate失活deep bed filtration 深层过滤defect structure 结构缺陷demonstrate论证，证明，证实；说明，表明，显示density 密度density difference密度差deposit沉积物design engineer设计师detector检测器detrimental有害的devastate破坏，蹂躏discolor变色，脱色drop in pressure 压降drying gas干燥气体drying干燥durability 耐久性，寿命，使用期限，强度eddy diffusion涡流扩散efficiency improvement 效率提高electromagnetic wave 电磁波Empirical model 经验模型employ采纳，利用end-of-pipe solution 最终方案energy efficiency 能量效率environmental health engineering环境健康工程equimolecular counter-diffusion 等分子反向扩散essential oil香精油ether 乙醚ethylene epoxidation 乙烯环氧化反应exothermic（放热的，endothermic吸热的,adiabatic绝热的）exotic chemistry 奇异化学explosive炸药extract 萃取液extract萃取液eddy 尾流，涡流electrical conductivity 导电性emission释放物，排放物emulsifying agent乳化剂energy saving 节约能量enthalpy 焓extract phase萃取相environmentally benign processing环境友好加工equilibrium constant 平衡常数ethyl alcohol乙醇)energy conversion 能量转化falling-rate降速falling-rate period降速干燥阶段feedstock 进料，原料fermentation发酵Fick’s Law 费克定律filter cake 滤饼filter paper 滤纸filtering medium 过滤介质filtering surface 过滤表面filtrate 滤液flux（通量，流通量）foodstuff 食品forces of friction 摩擦力forefront最前线，最前沿fouling污垢，发泡foundation 基础fraction collector馏分收集器fractional crystallization分步结晶fully developed turbulent flow充分发展湍流functional group官能团furnace 火炉，燃烧器fused quartz 熔化的石英factor 因数，因子，系数，比例family族feed liquor 进料液fertilizer化肥fiber纤维filter cake 滤饼filter 过滤器filtration 过滤final-proposal决议Fine chemical 精细化工first-principles基本原理，基本规则fluid mechanics 流体力学fluorescence 荧光色，荧光fossil fuel化石燃料freeze drying冷冻干燥gas diffusivity气体扩散性，气体扩散系数gasoline汽油general service facilities公用工程gravity重力gas adsorption 吸收；Gas and Liquid Chromatography气相色谱与液相色谱gasoline汽油governmental regulation政府规定heat exchanger 换热器heat requirement热负荷heat-sensitive material 热敏性物质heavy gas oil重瓦斯油helium氦HF alkylation氰氟酸烷基化high-efficiency高效的high-fidelity高保真的high-performance高性能human-factors engineering人类与环境工程humidity湿度hydrodynamic model水力学模型hydrogen and methane oxidation 氢气和甲烷氧化反应hydroquinone 对苯二酚hard hat 安全帽hazard identification 危害辩识，health care保健hearing aid助听器artificial limb假肢heat exchange 热交换heats of solution and vaporization溶解热和汽化热homologous series同系物human medicine人体医学hydrocarbon naphthene环烷烃hydrogen cyanide氰化氢i feed tray进料板ibuprofen异丁苯丙酸ifetime寿命impurity杂质in the vicinity of 在…附近，靠近..,大约…，在…左右information processing 信息处理injection注射inorganic salt无机盐insulation 绝缘inter-phase mass transfer界相际间质量传递is proportional to 与…成比例ideal system 理想系统In principle从原理而言in the absence of无---存在inert 惰性物，不参加反应的物质inflection:折射intensity强度，程度is almost inversely proportional to 几乎与…成反比internal structure 内部结构ketone酮kilogram千克labor 劳动力laminar sub-layer 层流底层latent heat潜热law 定律Lewis acid不可再生的路易斯酸liquid mixture 液体混合物lubricating oil润滑油Latin or Greek stem 拉丁或者希腊词根least-squares-regression最小二乘法life scientist生命科学家liquid-liquid extraction 液液萃取locant位次，位标loss prevention损失预防macroscopic phenomenon 宏观现象thermal stability热稳定性maintenance expense 维修费macroscopic particle 宏观的粒子materials science材料科学magnet 磁铁，有吸引力的人或物matte无光泽的，无光的mathematical expression steady-state model稳态模型mean value平均值m aximum最大的mechanical disturbance 机械扰动mechanical, thermal, chemical, electric, magnetic, and optical behavior. （机械性能、热学性能、化学性能、电学性能、磁性能、光学性能）mechanical separation 机械分离）medical electronics医疗电子medical instrumentation医疗器械medical engineering医学工程，医疗工程membrane technology膜技术medium 介质metal wall of a tube 金属管壁membrane module膜组件metallic solid 金属固体methane甲烷metallurgy 冶金学，冶金术milliliter毫升micro-reactor 微型反应器miscible可混合的，可溶的，可搅拌的mobile phase移动相model identification模式识别moisture content湿含量molecular diffusion 分子扩散motion of unbound electrons 自由电子的运动molecular transfer分子传递mount 安装，固定moving gas stream移动的气流multi-component distillation多组分精馏natural gas天然气naphtha石脑油near toequilibrium接近平衡natural and forced convection 自然对流和强制对流neural network神经网络net flow 净流量nitric acid 硝酸nitric acid硝酸nylon尼龙nitrogen oxides氮氧化物nonlinear-equation-solving technique非线性方程求解技术nomenclature of chemical compound化学化合物的命名法nuclear power 核能octane number of gasoline汽油辛烷值of the same order具有同一数量级of enthalpy 焓通量oil drilling采油operating condition操作条件oil shale油页岩overhead vapor塔顶汽体one-pass（单程）oxygenate content 氧含量operating cost 操作费用or convex（凹的，凹面）organic有机的，有机物的outline描绘，勾勒output产出，输出，产量oxides of nitrogen 氮氧化物paraffin石蜡，烷烃parent母链，主链penicillin青霉素partial vaporization 部分汽化particle size 颗粒尺寸pharmaceutical intermediate药物中间体pharmaceutical 制药packed tower 填料塔phaseout消除packing characteristics填充性质phenomenological model 现象模型packing material 填料pump 泵physical process 物理过程paint涂料physiologists生理学paraffin石蜡，链烷烃physiologist生理学家per unit area单位面积plastic塑料percent conversion百分比转化率plate tower 板式塔installation 装置feed 进料perpendicular to：与----垂直polymers聚合物Pharmaceutical制药power 动力phosgene synthesis 光气合成.precipitate 沉淀物plane chromatography薄层色谱precipitation沉淀，沉析plant layout工厂布局pressure gradient plant location工厂选址probability of failure失效概率pollutant 污染物process data historian：过程数据历史编撰师poor conductor of electricity 不良导电体process engineering过程工程porous medium 多孔介质process material过程物料(相对最终产品而言的）pretreatment 预处理process-simulation software packages过程模拟软件包process control（过程控制）product 产物，产品production line生产线prosthetics假肢器官学protein蛋白质psychology心理学pyridine砒啶purify 精制提纯qualitative定性的quantitative precision定量的精确quantitative relation定量关系radiation 辐射raffinate萃余液ratio of A to B A与B的比值rate of diffusion扩散速率reactant 反应物reaction 反应separation 分离reaction byproduct 反应副产物reaction yield反应产率reaction speed反应速度reaction zone反应区reactive distillation 反应精馏reactive distillation 反应精馏reactor energy input能量输入recycle 循环回收reboiler再沸器vaporization汽化regression model回归模型.refix前缀release释放，排放reflect 反射，replacement organ器官移植reflux回流replication 复制reforming重整resistance 阻力regenerate再生resistance 阻力，阻止replicating prefix重复前缀词retention volume保留体积retention times保留时间ring-shaped（环状的）Reynolds number雷诺准数rules and regulations 规章制度reaction kinetics 反应动力学reactant 反应物retardation factor保留因子，延迟因子safety experience安全经验safety knowledge安全知识safety management support安全管理基础知识safety shoe防护鞋sample样品sandstone砂岩sedimentary rock沉积岩sanitation卫生segment段，片，区间，部门，部分；弓形，圆缺；分割，切断screening筛选，浮选sensor 传感器，探头seepage渗出物siltstone泥岩self-taught自学size distribution 粒度分布semiconductor 半导体sodium hydroxide 氢氧化钠sensible heat（sensible heat：显热）solubility，溶解度，溶解性separation of solids 固体分离solute溶质settling tank沉降槽solution溶液setup 装置solvent make-up 补充溶剂optimum 最优的shape 形状solvent 溶剂s oap肥皂solvent-recovery system 溶剂回收系统solid state physics固体物理学stage-type distillation column级板式精馏塔soluble可溶解的solvent溶剂statistical technique 统计技术specialized group专业组storage仓库specialty chemical特殊化学品，特种化学品stripping section汽提段,提馏段spinning electrons 旋转电子structural steels 结构钢spray chamber 喷淋室substituent取代基stationary nonvolatile phase静止的不挥发相substitute取代，替代steam蒸汽suffix后缀stereoselective立体选择性的surface layer 表面层straight line：直线surface treatment表面处理streptomycin（链霉素）surgeon外科医生styrene苯乙烯suspension 悬浮液sublime升华supportive or substitute organ辅助或替代器官synthetic rubber合成橡胶tanker油轮kerosene煤油tarry柏油的，焦油的，焦油状的technical competence技术能力tangible return有形回报terminology术语，词汇tar sand沥青石thermal conductivity 导热性technical advance 技术进步thermal diffusion热扩散technical challenge技术挑战，技术困难total condense全凝器technical evaluation技术评估tower shell 塔体Telecommunication 电信toxic有毒的temperature gradient 温度梯度transport of momentum 动量传递temperature gradient 温度梯度tray 塔板temperature level 温度高低turbulent flow 湍流test mixture测试混合物two-film theory 双膜理论the Chemical Manufacturers Association化工生产协会two-phase flow两相流the random motion of molecules 分子无规则运动thermal conduction 热传导thermodynamic equilibrium热力学平衡tonnage吨位，吨数，吨产量top 顶部，塔顶trap 收集，捕集triple bond三健，三价unconverted reactant未转化的反应物unabsorbed component未吸收组分purity纯度urea尿素vacant atomic site 原子空位vacuum drying真空干燥vapor-liquid contacting device汽液接触装置Vacuum 真空viscous resistance粘性阻力v alve tray浮阀塔板volatility挥发性vapor pressure蒸汽压voluntary自愿的，无偿的，义务的；有意的，随意的；民办的via经，由，通过，借助于viscosity 黏度waste 废物wastewater treatment污水处理waste disposal废物处理water droplet水珠，水滴water purification水净化wax石蜡weir溢流堰with the result that：由于的缘故，鉴于的结果water-cooling tower水冷塔yield 产率，收率。

自动控制原理重点英文单词+汉语翻译

Closed-loop control systems 闭环控制系统Open-loop control systems开环控制系统Process 过程Linear system线性系统Nonlinear system非线性系统Continuous system 连续系统Discrete system离散系统Stability 稳定性Steady-state performance 稳态性能Transient performance 暂态特性mathematical model 数学模型differential equations 微分方程Transfer function传递函数zeros and poles of transfer function传递函数的零极点Inverse proportion part 比例环节Inertia part惯性环节Integral part 积分环节Derivative part 微分环节Vibrate part震荡环节Delay part 滞后环节Block diagram (Equivalent transformation)方框图(Unit) negative (positive) feedback loop 负（正）反馈回路Mason formula 梅逊公式disturbance 干扰Step signal 阶跃信号Ramp signal(speed function signal) 斜波信号Parabola signal(acceleration signal) 加速度信号pulse signal脉冲信号Sinusoidal signal 正弦信号Delay time延迟时间Rise time 上升时间Peak time峰值时间Settling time 稳定时间Percent overshoot超调量Steady-state error稳态误差position error coefficient Kp speed error coefficient Kv acceleration error coefficient Kafirst-order system一阶系统S econd-order system 二阶系统high-order system 高阶系统Dominant pole主导极点Underdamped欠阻尼Critically Damped临界阻尼Overdamped 过阻尼Undamped无阻尼的Routh-Hurwitz stability criterion劳斯稳定性判据R outh array 劳斯表Character equation 特征方程root locus 根轨迹open-loop zeros and poles 开环零极点Magnitude and angle requirements of root locus幅值与相角frequency character 频率特征(inverse) Laplace transformation 拉普拉斯（反）变换Nyquist plot奈奎斯特图Bode diagram波德图Logarithmic magnitude frequency character对数幅值频率特性Logarithmic phase frequency character对数相频特征Nyquist stability criterion奈奎斯特稳定判据cutoff frequency 剪切频率Phase margin 相位裕量Gain margin 增益裕量Cutoff frequencyCascade phase-lead compensation串联超前矫正Cascade phase-lag compensation 串联滞后校正Cascade phase-lag and -lead compensation串联滞后-超前矫正sample control system 采样控制系统digital control system 数控系统discrete control system离散控制系统Shannon sampling theorem 香农采样定理Zero-order hold 零阶保持sampling period 采样周期Sampling frequency 采样频率Z-transform z变换Z-inverse transform z逆变换pulse transfer function脉冲传递函数bilinear transform双线性变换。

generalized harmonic modeling

generalized harmonic modeling介绍如下：Generalized Harmonic Modeling (GHM) is a technique used to accurately predict the behavior of electrical power systems with non-linear loads. It is a mathematical modeling method that uses sinusoidal functions with different frequencies and harmonics to represent the voltage and current waveforms of complex electrical systems.GHM is used to analyze and simulate power systems with non-linear loads, such as computer equipment and power electronic devices. These loads cause high levels of harmonic distortion, which can negatively impact the efficiency, reliability, and safety of the power system.GHM models the impedance of the non-linear loads using harmonic currents and voltages, and then uses frequency-domain analysis to obtain the steady-state voltage and current waveforms. The model includes all the harmonic components of the voltage and current signals, allowing for accurate prediction of distortion levels in the power system.GHM is commonly used in power system analysis and design to optimize the performance of electrical grids and ensure the safe and reliable supply of power to end-users. It is especially useful in high-voltage power systems, where harmonic distortion can cause significant operational problems.。

英汉数学物理常用词汇

Cauchy's integral theorem，科西积分定理
Cauchy's integral formula，科西积分公式
Liouville's theorem，刘维尔定理
Sequence，序列
sequence of number，数列
sequence of function，函数列
derivative，导数
partial derivative，偏导数
higher derivative，高阶导数
derivative of order n/nth order derivative，n阶导数
analytic function，解析函数
regular function，正则函数
transcendented function，超越函数
hyperbolic function，双曲函数
anti-hyperbolic function，反双曲函数
trigonometric function，三角函数
anti-trigonometric function，反三角函数
analyticity，解析性
general term，通项
series，级数
series of numbers，数项级数
series of functions/functional series，函项级数
number of term，项数
infinite series，无穷级数
partial sum，部分和
remainder term，余项
multiply valued function，多值函数
simple function，单叶函数

力学专业英语词组+解释

拉力 tensile force正应力 normal stress切应力 shear stress静水压力 hydrostatic pressure集中力 concentrated force分布力 distributed force线性应力应变关系 linear relationship between stress andstrain弹性模量 modulus of elasticity横向力 lateral force transverse force轴向力 axial force拉应力 tensile stress压应力 compressive stress平衡方程 equilibrium equation静力学方程 equations of static比例极限 proportional limit应力应变曲线 stress-strain curve拉伸实验 tensile test‘屈服应力 yield stress极限应力 ultimate stress轴 shaft梁 beam纯剪切 pure shear横截面积 cross-sectional area挠度曲线 deflection curve曲率半径 radius of curvature曲率半径的倒数 reciprocal of radius of curvature纵轴 longitudinal axis悬臂梁 cantilever beam简支梁 simply supported beam微分方程 differential equation惯性矩 moment of inertia静矩 static moment扭矩 torque moment弯矩 bending moment弯矩对x的导数 derivative of bending moment with respect to x弯矩对x的二阶导数 the second derivative of bending moment with respect to x 静定梁 statically determinate beam静不定梁 statically indeterminate beam相容方程 compatibility equation补充方程 complementary equation中性轴 neutral axis圆截面 circular cross section两端作用扭矩 twisted by couples at two ends刚体 rigid body扭转角 twist angle静力等效 statically equivalent相互垂直平面 mutually perpendicular planes通过截面形心 through the centroid of the cross section 一端铰支 pin support at one end一端固定 fixed at one end弯矩图 bending moment diagram剪力图 shear force diagram剪力突变 abrupt change in shear force、旋转和平移 rotation and translation虎克定律 hook’s law边界条件 boundary condition初始位置 initial position、力矩面积法 moment-area method绕纵轴转动 rotate about a longitudinal axis横坐标 abscissa扭转刚度 torsional rigidity拉伸刚度 tensile rigidity剪应力的合力 resultant of shear stress正应力的大小 magnitude of normal stress脆性破坏 brittle fail对称平面 symmetry plane刚体的平衡 equilibrium of rigid body约束力 constraint force重力 gravitational force实际作用力 actual force三维力系 three-dimentional force system合力矩 resultant moment标量方程 scalar equation、矢量方程 vector equation张量方程 tensor equation汇交力系 cocurrent system of forces任意一点 an arbitrary point合矢量 resultant vector反作用力 reaction force反作用力偶 reaction couple转动约束 restriction against rotation平动约束 restriction against translation运动的趋势 tendency of motion绕给定轴转动 rotate about a specific axis沿一个方向运动 move in a direction控制方程 control equation共线力 collinear forces平面力系 planar force system一束光 a beam of light未知反力 unknown reaction forces参考框架 frame of reference大小和方向 magnitude and direction几何约束 geometric restriction刚性连接 rigidly connected运动学关系 kinematical relations运动的合成 superposition of movement固定点 fixed point平动的叠加 superposition of translation刚体的角速度 angular speed of a rigid body质点动力学 particle dynamics运动微分方程 differential equation of motion工程实际问题 practical engineering problems变化率 rate of change动量守恒 conservation of linear momentum定性的描述 qualitative description点线 dotted line划线 dashed line实线 solid line矢量积 vector product点积 dot product极惯性矩 polar moment of inertia角速度 angular velocity角加速度 angular accelerationinfinitesimal amount 无穷小量definite integral 定积分a certain interval of time 某一时间段kinetic energy 动能conservative force 保守力damping force 阻尼力coefficient of damping 阻尼系数free vibration 自由振动periodic disturbance 周期性扰动viscous force 粘性力forced vibration 强迫震动general solution 通解particular solution 特解transient solution 瞬态解steady state solution 稳态解second order partial differential equation 二阶偏微分方程external force 外力internal force 内力stress component 应力分量state of stress 应力状态coordinate axes 坐标系conditions of equilibrium 平衡条件body force 体力continuum mechanics 连续介质力学displacement component 位移分量additional restrictions 附加约束compatibility conditions 相容条件mathematical formulations 数学公式isotropic material 各向同性材料sufficient small 充分小state of strain 应变状态unit matrix 单位矩阵dilatation strain 膨胀应变the first strain invariant 第一应变不变量deviator stress components 应力偏量分量the first invariant of stress tensor 应力张量的第一不变量bulk modulus 体积模量constitutive relations 本构关系linear elastic material 线弹性材料mathematical derivation 数学推导a state of static equilibrium 静力平衡状态Newton‘s first law of motion 牛顿第一运动定律directly proportional to 与……成正比stress concentration factor 应力集中系数state of loading 载荷状态st venant’ principle 圣维南原理uniaxial tension 单轴拉伸cylindrical coordinates 柱坐标buckling of columns 柱的屈曲critical value 临界值stable equilibrium 稳态平衡unstable equilibrium condition 不稳定平衡条件critical load 临界载荷a slender column 细长杆fixed at the lower end 下端固定free at the upper end 上端自由critical buckling load 临界屈曲载荷potential energy 势能fixed at both ends 两端固定hinged at both ends 两端铰支tubular member 管型杆件transverse dimention 横向尺寸stability of column 柱的稳定axial force 轴向力elliptical hole 椭圆孔plane stress 平面应力nominal stress 名义应为principal stress directions 主应力方向axial compression 轴向压缩dynamic loading 动载荷dynamic problem 动力学问题inertia force 惯性力resonance vibration 谐振static states of stress 静态应力dynamic response 动力响应time of contact 接触时间length of wave 波长resonance frequency 谐振频率。

The Method of Moments for Higher Moments-- Problems, Solutions, and Tools

0
Abstract1 Nhomakorabea1 Introduction
The Method of Moments (MoM) can be described as follows: a typical, tagged customer is traced as it proceeds through the system and the components of its delay are analyzed separately. Taking the moments and the probabilities for the delay components, moments for the overall delay can be derived. Usually, the only assumptions about the model are that it is in steady state, that the waiting line has in nite capacity, and that interarrival times are exponentially distributed. Cobham invented this technique in 1954, when he derived expected waiting times in M/GI/1{ PRIO NP] queueing systems 1]. Its importance is re ected in a contemporary cite by Phipps: "In his article Cobham demonstrated that (although probability distributions connected with waiting-line problems, except in simple cases, continue to defy analysis because of ingrained nonMarkovian probability relations) certain expected value characteristic of the waiting process can be easily calculated." 12]. We can hope to derive moment information, though the process as a whole cannot be described and solved, and though the sojourn and/or waiting time distribution or its transform cannot be derived. This is important to note, because moments can be obtained by di erentiation of the corresponding transform (see 8] and Section 4 of the paper). Nowadays, the MoM, though mostly not explicitly mentioned, is one of the most important techniques in queueing analysis. The fate of tagged customers is frequently used to analyze polling and vacation as well as priority and feedback systems. Mostly, the MoM is applied as a straightforward and simple way to obtain rst moment information in complex M/GI/1 queueing systems. To our knowledge, no real attempt has been made to derive higher moments by using the MoM. Though the complexity can be expected to be much higher, the MoM for higher moments might be a tool as powerful as for the rst moment analysis. Up to now, only Herzog applied the MoM for a second moment analysis in the M/M/1-FCFS system 7]. Since he heavily uses the memoryless property of the service time process, a more general study is required. In this paper, we evaluate the suitability of the MoM for higher moments. In the next section, we give a generalized view on the mathematical nature of the MoM. When applying this general technique to the second moment analysis of an M/GI/1{FCFS system, we run into considerable mathematical problems. After these problems have been solved in Section 3, we return to the M/GI/1{FCFS system and present a solution in Section 4. In Section 6, we resume our experience with the case study, give nal comments on the suitability of the MoM for the analysis of higher moments, and emphasize on the tools we applied to deal with the inherent complexity of the method.

系统的能控性和能观性英文版

Unit 13 Controllability and ObservabilityA system is said to be controllable at time 0t if it is possible by means of an unconstrained control vector to transfer the system from any initial state )(0t x to any other state in a finite interval of time. A system is said to be observable at time 0t if, with the system in state )(0t x , it is possible to determine this state from the observation of the output over a finite time interval.The concepts of the controllability and observability were introduced by Kalman. They play an important role in the design of control systems in state space. In fact, the conditions of controllability and observability may govern the existence of a complete solution of the control system design problem. The solution to this problem may not exist of the system considered is not c ontrollable. Although most physic al systems are c o ntrollable and observable, corresponding mathematical models may not possess the property of controllability and observability.Complete State Controllability of Continuous-Time SystemsConsider the continuous-time systemBu AX X+= (13. 1) where X=state vector (n -vector)u =control signal (scalar) A=n n ⨯ matrix B=1⨯n matrixThe system described by Equation (13. 1) is said to be state controllable at 0t t =if it is possible to construct an unconstrained control signal that will transfer an initial state to any final state in a finite time interval 10t t t ≤≤. If every state is controllable, then the system is said to be completely state controllable.We shall now derive the condition for complete state of controllability. Without loss of generality, we can assume that the final state is the origin of the state space and that the initial time is zero,or 00=t .The solution of Equation (13. 1) is⎰-+=tt A Atd Bu eX e t X 0)()()0()(τττApplying the definition of complete state controllability just given, we have⎰-+==111)(1)()0(0)(t t A At d Bu eX et X τττor⎰--=10)()0(t A d Bu eX τττ(13. 2)And τA e -can be written∑-=-=1)(n k kkA A e τατ(13. 3)Substituting Equation (13. 3) into Equation (13. 2) gives∑⎰-=-=101)()()0(n k t k kd u B A X τττα (13. 4)Let us put⎰=1)()(t k k d u βττταThen Equation (13. 4) becomes∑-=-=1)0(n k k kB A X β[]⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=--1101n n B A AB Bβββ(13. 5) If the system os completely state controllable, then, given any initial state X(0), Equation(13. 5) must be satisfied. This requires that the rank of the n n ⨯matrix[]B AAB Bn 1-be n .From this analysis, we can state the condition for complete state controllability as follow s. The system given by Equation (13. 5) is completely state controllable if and only if the vectorsB AAB B n 1,- are linearly independent, or the n n ⨯matrix[]B AAB Bn 1-is the rank n.The result just obtained can be extended to the case where the control vector U is r-dimensional. If the system is described byBU AX X+= Where U is an r -vector, then it can be proved that the condition of for complete statecontrollability is that the n n ⨯matrix[]B AAB B n 1-be of rank n , or contain n linearly independent column vectors. The matrix[]B AABBn 1-is commonly called the controllability matrix.Complete Observability of Continuous-Time SystemIn this section we discuss the observability of linear systems. Consider the unforced system described by the following equationsAX X= (13. 6) CX Y = (13. 7)where X=state vector (n -vector)Y=output vector (m -vector) A=n n ⨯matrix C=n m ⨯matrixThe system is said to be completely observable if every state )(0t X can be determined from the observation of Y(t) over a finite time interval,10t t t ≤≤. The system is, therefore, completely observable if every transition of the state eventually affects every element of the output vector. The concept of observability os useful in solving the problem or reconstructing unmeasurable state variable from measurable variables in the minimum possible length of time. In this section we treat only linear, time-invariant systems. Therefore, without loss of generality, we can assume that 00=t .The concept of observability is very important because, in practice, the difficulty encountered with state feedback control is that some of the state variables are not accessible for direct measurement, with the result that it becomes necessary to estimate the unmeasurable state variables in order to construct the control signals.● Such estimates of state variables are possible of and only if the system is completely observable.In discussion observability conditions, we consider the unforced system as given by Equation (13. 6) and (13. 7). The reasons for this are as follows, If the system is described byBu AX X+= Bu CX Y +=then⎰-+=tt A Atd Bu eX e t X 0)()()0()(τττAnd Y(t) is⎰++=-tt A AtDu d Bu eC X Cet Y 0)()()0()(τττSince the matrices A, B, C, and D are known and u(t) is also known,the last terms onthe right-hand side of this last equation are known quantities. Therefore, they may be subtracted from the observed value of Y(t). Hence, for investigating a necessary and sufficient condition for complete observability, it suffices to consider the system described by Equations (13. 6) and (13. 7).Consider the system described by Equations (13. 6) and (13. 7). The output vector Y(t) is)0()(X Cet Y At=And At e can be written as∑-==1)(n k kkAtA t e αHence, we obtain∑-==1)0()()(n t kkX CA t t Y αor)0()()0()()0()()(1110X CAt CAX t CX t t Y n n --+++=ααα (13. 8)If the system is completely observable, then, given the output Y(t) over a time interval ≤≤t t 0 1t , X(0)is uniquely determined from Equation (13. 8). It can be shown that this requires therank of the n nm ⨯matrix⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡-1n CA CA C to be n.From this analysis we can state the condition for complete observability as follows.The system described by Equation (13. 6) and (13. 7) is completely observable of and only is the n nm ⨯matrix⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡-1n CA CA C is of rank n or has n linearly independent column vectors. This matrix is called the observability matrix.Key Words and Terms1. controllability n. 可控性2. observability n. 可观测性3. controllable adj. 可控的4. observable adj. 可观测的5. mathematical model 数学模型6. property n. 性质，属性7. continuous-time system 连续时间系统 8. generality n. 一般性，普遍性 9. rank n. 秩10. linearly independent 线性无关 11. time-invariant system 时变系统 12. suffice v. 满足NotesAlthough most physic al systems are controllable an d observable, corresponding mathematical models may not possess the property of controllability and observability.尽管大多数的物理系统都是可控的和可观测的，它们所对应的数学模型并不一定具有可控性和可观测性。

稳态近似法和平衡假设法

稳态近似法和平衡假设法英文回答：Steady State Approximation (SSA)。

The steady state approximation (SSA) is a mathematical technique used to simplify the analysis of stochastic systems. It assumes that the system is in a steady state, where the probability distribution of the system remains constant over time. This assumption can be made when the system is in equilibrium or when the changes in the system are relatively slow compared to the rate at which the system adjusts to those changes.The SSA can be applied to a wide range of systems, including queuing networks, inventory systems, and ecological systems. In queuing networks, the SSA can be used to approximate the distribution of the number of customers in the system. In inventory systems, the SSA can be used to approximate the distribution of the inventorylevel. In ecological systems, the SSA can be used to approximate the distribution of the population size.The SSA is a powerful tool that can be used to simplify the analysis of stochastic systems. However, it is important to note that the SSA is an approximation and may not be accurate in all cases.Equilibrium Assumption.The equilibrium assumption is a special case of the SSA that assumes that the system is in a state of equilibrium. In equilibrium, the probability distribution of the system does not change over time. This assumption can be made when the system is not subject to any external disturbances.The equilibrium assumption can be used to simplify the analysis of a wide range of systems, including physical systems, chemical systems, and biological systems. In physical systems, the equilibrium assumption can be used to determine the equilibrium position of a particle or object. In chemical systems, the equilibrium assumption can be usedto determine the equilibrium concentrations of reactantsand products. In biological systems, the equilibrium assumption can be used to determine the equilibrium distribution of species in an ecosystem.The equilibrium assumption is a powerful tool that can be used to simplify the analysis of a wide range of systems. However, it is important to note that the equilibrium assumption is an approximation and may not be accurate inall cases.中文回答：稳态近似法。

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A general steady state mathematical model for ﬁn-and-tube heatexchanger based on graph theoryJian Liu a ,WenJian Wei a ,GouLiang Ding a,*,Chunlu Zhang a ,Masaharu Fukaya b ,Kaijian Wang b ,Takefumi Inagaki baDepartment of Power and Energy Engineering,Institute of Refrigeration and Cryogenics,Shanghai Jiaotong University,1954Huashan Road,Shanghai 200030,ChinabFujitsu General Institute of Air-Conditioning Technology Limited,1116Suenage,Takatsu-Ku,Kawasaki 213-8502,JapanReceived 7January 2004;received in revised form 25June 2004;accepted 25June 2004AbstractFin-and-tube heat exchangers are widely used in air conditioners,chillers,etc.A lot of factors,including arrangement of refrigerant circuits,conﬁgure speciﬁcation of ﬁns and tubes,and operating conditions,have signiﬁcant inﬂuence on the performance of ﬁn-and-tube heat exchangers.For the purpose of fast design of high performance heat exchangers,a simulator reﬂecting the inﬂuence of these factors is necessary.In this paper,a general steady state mathematic model based on the graph theory is presented.With the help of the directed graph and graph-based traversal methods (Breadth-ﬁrst search and Depth-ﬁrst search),this model is capable to describe any ﬂexible refrigerant circuit arrangement,and quantify the refrigerant distribution in the refrigerant circuit and heat conduction through ﬁns.An alternative iteration method is also developed to solve the conservation equations,which can shorten the simulating time effectively.The model is veriﬁed with the experimental results,and the maximum error is within G 10.0%.A simulator based on this model has been used for designing practical ﬁn-and-tube heat exchangers.q 2004Elsevier Ltd and IIR.All rights reserved.Keywords:Heat exchanger;Finned tube;Modelling;Heat transfer;Steady state;Pressure dropMode`le mathe ´matique du re ´gime permanent d’un e ´changeur de chaleur a`tubes ailete ´s fonde ´sur la the ´orie graphique Mots cle´s:Echanger de chaleur;Tube ailete ´;Mode ´lisation;Transfert de chaleur;Re ´gime permanent;Chute de pression 1.IntroductionFin-and-tube heat exchangers are widely used in airconditioners,chillers,etc.A lot of factors,including arrangement of refrigerant circuits,conﬁgure speciﬁcation of ﬁns and tubes,and operating conditions,have signiﬁcant inﬂuence on the performance of ﬁn-and-tube heat0140-7007/$35.00q 2004Elsevier Ltd and IIR.All rights reserved.doi:10.1016/j.ijrefrig.2004.06.008*Corresponding author.Tel.:C 86-21-62932110;fax:C 86-21-62932601.E-mail address:glding@ (G.L.Ding).exchangers.For the purpose of fast design of high performance heat exchangers,a simulator reﬂecting the inﬂuence of these factors is necessary.Unfortunately,it is not easy to develop a model for such kind of simulator because of two main difﬁculties.(1)The methods used to arrange the refrigerant circuits are almost unlimited in practical design.It is difﬁcult toﬁnd a simple way to describe all of the possible refrigerant circuits.(2) The simulating time consumption used to solve the coupled conversation equations in a distributed parameters are almost intolerance in practical design.It is difﬁcult to develop an effective algorithm to solve those coupled conversation equations in a short time.Many researchers have developed distributed parameter models to analyse the steady state performance ofﬁn-and-tube heat exchangers.However,most of the models are only suitable forﬁn-and-tube heat exchangers with simple tube arrangement,such as X.Jia et al.(1999)[1],Judge and Radermacher(1997)[2],Theerakulpisut et al.(1998)[3]. One method that is capable of analyzing the performance of heat exchanger with complex refrigerant circuitry is tube-by-tube technology developed by Domanski(1991)[4]. This method was further developed by Lee et al.(2002)[5] to study two-dimension air distribution.However,both of them did not consider the effect of the heat conduction throughﬁns.Liang et al.(2001)[6]used a distributed model to analyze the performance of heat exchangers with complex refrigerant circuitry,but no detailed algorithm was presented.So a general model and corresponding effective algorithm used to evaluate the performance is still lacking,and further study is needed.This study attempts to construct a general steady state mathematical model that can describe any refrigerant circuits,and evaluate the effects of refrigerant distribution and the heat conduction in aﬁn-and-tube heat exchanger.A novel algorithm is also developed to solve the conservation equations and shorten simulating time.2.Mathematical modelA typicalﬁn-and-tube heat exchanger generally consists of corrugated or plainﬁn plates assembled over a bank of coils.A distributed parameter model is used to analyze the detailed local behaviors and understand the mechanism of heat and mass transfer,and the whole heat exchanger is divided into several control volumes.Each control volume contains three parts:the refrigerant inside tube,ﬁn-and-tube, and the air outside tube.2.1.Description of tube connectionIn order to accurately describe the complex refrigerant circuits,an adjacency graph and corresponding adjacent matrix in the graph theory is introduced to describe the connection among each tube,and trace the conﬂuence and division of refrigerantﬂow.The directed graph is a kind of conceptualized hierarchy, depicted as a set of vertices connected by edges and each edge is endowed with certain direction.As far as heat exchanger considered,one vertex denotes a heat exchange tube,and the edge denotes the relationship between two tubes.Because the refrigerantﬂow direction has been considered,the edge is also endowed withﬂow direction.NomenclatureA o Heat transfer area on air side(m2)A i Inside surface area of tube(m2)D i Inner diameter of tube(m)D o Outer diameter of tube(m)m Massﬂow rate(kg s K1)f Friction factorG Massﬂux(kg m K2s K1)h Speciﬁc enthalpy(kJ kg K1)L Length(m)P Pressure(Pa)Q Heat exchange(W)T Temperature(K)x QualityGreekD p Pressure drop(Pa)a Heat transfer coefﬁcient(kW m K2K K1) r Density(kg m K3)3Void fractionh0Fin surface efﬁciency Subscriptsa Airacc Accelerationback Backbottom Bottomf Frictionﬁnfront Frontin Inletl Liquido Outletr Refrigeranttop Toptot Totalv Vaporwall Tube wallJ.Liu et al./International Journal of Refrigeration27(2004)965–973 966Thus the kind of adjacent graph can be used to describe any ﬂexible refrigerant circuits.The adjacency matrix is a mathematic data structure used to express above directed graph.The value of matrix element is expressed as follows:m i ;j Z0when No :j is not connected to No :i ;the value is 01when No :j is connected to No :i ;the value is 1(According to the adjacent matrix,the simulator will easily get the information of conﬂuence and division in each tube,and trace the refrigerant ﬂow direction.For example,a simpliﬁed heat exchanger with complex refrigerant circuitry and typical control volume are shown as Fig.1,in which 2rows and 4columns tubes are assembled and the refrigerant ﬂow splits into two branches.In order toidentify each tube in the heat exchanger,each tube is coded in order from ﬁrst row to last row (shown in Fig.1).There are two additional artiﬁcial tubes added in #0and #9,which represent the inlet refrigerant header and outlet refrigerant header,respectively.According to the heat exchanger in Fig.1,the corresponding directed graph and adjacent matrix M can be obtained,which are presented in Figs.2and 3,respectively.2.2.Equations of control volume2.2.1.AssumptionOn the basis of the analysis of actual operating conditions of heat exchanger,the following assumptions are applied:(1)The refrigerant ﬂow inside the tube is one-dimensionalaxialﬂow.Fig.2.The directed graph corresponding to Fig.1.Fig.1.Schematic diagram of a simpliﬁed heat exchanger and a single control volume.J.Liu et al./International Journal of Refrigeration 27(2004)965–973967(2)The heat transfer along tube length direction isneglected.(3)The pressure drop on airside is neglected because it isusually very small.erning equations on refrigerant sideEnergy equation for refrigerant:Q1r Z Q2r(1)where,Q1r and Q2r represent the heat exchange by enthalpy difference and that by temperature difference,respectively, calculated as follows:Q1r Z m rðh r;in K h r;outÞ(2)Q2r Z a r A iT r;in C T r;out2K T wall(3)Here,m r is refrigerant massﬂow rate;h r,in and h r,out are inlet and outlet refrigerant speciﬁc enthalpies of the control volume,respectively;T r,in and T r,out are inlet and outlet refrigerant temperatures of the control volume,respectively. The heat transfer coefﬁcient a r is calculated by empirical correlation.Momentum equation for refrigerant:D p total Z D p f C D p acc(4) where,D p total,D p acc and D p f represent total pressure drop, acceleration pressure drop and friction pressure drop of refrigerant,respectively.In two-phase region,the accelera-tion pressure drop is calculated as follows:D p acc Z m2rx2r;outr v3r;outCð1K x r;outÞ2r lð1K a r;outÞK m2rx2r;inr v3r;inCð1K x r;inÞ2r lð1K a r;inÞ(5)2.2.erning equations on airsideEnergy equation for mainstream air:Q1a Z Q2a(6) where,Q1a and Q2a represent heat exchange by enthalpy difference and that by temperature difference,Q1a Z m aðh a;in K h a;outÞ(7)Q2a Z a a A o h oT a;in C T a;out2K T wall(8)Here,air massﬂow rate m a is obtained based on upstream control volumes in front row;h a,in and h a,out are inlet and outlet air speciﬁc enthalpies of the control volume, respectively;T a,in and T a,out are inlet and outlet air dry bulb temperatures of the control volume,respectively.The heat transfer coefﬁcient a a is calculated by empirical erning equations forﬁn-and-tubeConsidering the heat conduction throughﬁns,the energy equation is expressed as follows:Q1r C Q1a C Q cond Z0(9)where,Q cond is the total heat conduction byﬁns,calculated as follows:Q cond Z Q front C Q back C Q top C Q bottom(10) where,Q front,Q back,Q top,and Q bottom are heat conductions throughﬁns from front row,back row,upper column,and bottom column,respectively.They are calculated by the temperature difference between the tube wall temperature of current control volume and corresponding temperature of neighbor tube(shown as Fig.1).Eqs.(1),(4),(6)and(9)are the whole governing equations for a typical control volume.2.2.5.Analyses on conservation equationsThe input parameters of a control volume are inlet refrigerant enthalpy h r,in,inlet refrigerant pressure p r,in, refrigerant massﬂow rate m r,and inlet air dry bulb temperature T db,in,inlet air wet bulb temperature T wb,in, and air massﬂow rate m a.Therefore,there areﬁve unknown variables(T a,out,T r,out,h r,out,h a,out,T wall).But when the air/refrigerant state equations are introduced,only three unknown variables(T a,out,h r,out,and T wall)are left,the set of Eqs.(1),(6),and(9)is solvable.The outlet refrigerant pressure drop can be obtained by Eq.(4).2.3.Conﬂuence and division of branchFor a heat exchanger with a complex refrigerant circuit, the governing equations at the division/conﬂuence points are needed to determine the inlet refrigerant state parameters of down stream branches.The following equations are adapted to branch con-ﬂuence,and other parameters can be obtained according to following two variables:h in ZP nk Z1h k;out m kP nk Z1k(11) p in Z/Z p k;out Z/Z p n;outðk Z1w nÞ(12)where,h in and p in represent the speciﬁc enthalpy and pressure of refrigerant after conﬂuence,respectively;h k,out, p k,out and m k are the speciﬁc enthalpy,pressure and mass ﬂow rate of refrigerant at the outlet of the‘k’sub-branch upstream,respectively;n is the number of conﬂuence branch.The following equations are adapted to branch division, and other parameters can be obtained according to following two variables:h out Z h1;in Z/Z h k;in Z/Z h n;inðk Z1w nÞ(13)J.Liu et al./International Journal of Refrigeration27(2004)965–973 968p out Z p1;in Z/Z p k;in Z/Z p n;inðk Z1w nÞ(14) where,h out and p out represent the speciﬁc enthalpy and pressure of refrigerant before division,respectively;h k,in and p k,in are the speciﬁc enthalpy and pressure of refrigerant at the inlet of the‘k’sub-branch downstream,respectively.2.4.Refrigerant distributionIn a heat exchanger with multiple circuits,refrigerant massﬂow rate will be adjusted automatically until the refrigerant pressure drops in all circuits from inlet to outlet are equal.Eq.(15)uses a simple expression to reﬂect the relationship between the refrigerant massﬂow rate m and the pressure drop D p according to Jung’s correlation.D p Z Sm2(15) where,S is the equivalentﬂow resistance for a given circuit branch.Initially,by assigning the refrigerant massﬂow rate in each branch,which starts from the same division point and ends at the outlet of heat exchanger,the refrigerant pressure drop D p i(i Z1,.,n)in each branch can be obtained, according to Eq.(4),then the value of S in each branch can be got:S1Z D p1m21;.;S k Z D p km2k;.;S n Z D p nm2n(16)where,S k,D p k,and m k are the equivalentﬂow resistance, pressure drop and refrigerant massﬂow rate of the‘k’branch(k Z1w n),respectively.The distribution of refrigerant massﬂow rate in each branch can be adjusted to ensure that the pressure drop of each branch is the same(D p1Z D p2Z.Z D p n)by using previous calculated equivalentﬂow resistance S i(i Z 1,.,n),so the ratio of refrigerant massﬂow rate in each branch is express as follows:m1:m2:/:m n Z1ﬃﬃﬃﬃﬃS1p:1ﬃﬃﬃﬃﬃS2p:/:1ﬃﬃﬃﬃﬃSnp(17)Since the sum of the refrigerant massﬂow rates is equal to the massﬂow rate at the division point,the ratio of the refrigerant massﬂow in the‘k’branch is calculated as:3k ZS K0:5kP nj Z1S j(18)where,3k is the ratio of the refrigerant massﬂow rate in the ‘k’branch to the total refrigerant massﬂow rate m total(k Z 1w n).By the Eq.(18)and total refrigerant massﬂow rate m total, refrigerant distribution in each branch can be determined.2.5.Heat transfer and pressure drop correlationsIn order to simulate the evaporation and condensation process,special empirical heat transfer and pressure drop correlations have been chosen,and listed as Table1.The properties of refrigerant are based on REFPROP Ver6.0 [17].3.AlgorithmAn overall alternating iterative algorithm based on graph-based traversal method is developed to solve the distributed parameter model.In this algorithm,the energy equations of all control volumes are solved simultaneously independent of the momentum equationsﬁrstly,and then the momentum equations of all control volumes are solved independent of energy equations.Therefore,the coupled relationship of energy conservation equations and momen-tum conservation equations is decoupled.Meanwhile,in order to realize above process,the basic concepts of Breadth-ﬁrst search(BFS)and Depth-ﬁrst search(DFS)in graph theory are introduced to create the appropriate searching path.By comparing with the previous algorithm, which is used to solve those coupled conservation equations, the iteration number has been largely reduced and it leads to sharply reduction of whole simulation time.The compu-tation time with the new method is only1/40to1/60of the previous one.This algorithm mainly consists of three parts,shown as Fig.4:(1)Creation of computation sequences,in which twodifferent calculation sequences,called heat transfer path and pressure drop path,are used to accomplish the solving process of energy conservations equations and that of momentum conservation equations.(2)Heat transfer calculation process,which solves theenergy conservation equations of all control volumes independent of the momentum equations along the heat transfer path.(3)Pressure drop calculation process,which solves themomentum equations of all control volumes indepen-dent of the energy equations along pressure drop path, and then adjusts the refrigerant massﬂow rate in each branch based.The solution is iterated until the overall heat conduction ofﬁn is less than speciﬁed criteria.The following presents the detailed contents of each part.3.1.Creation of computation sequences3.1.1.Heat transfer pathThe BFS is a basic search algorithm that is used to visit all the vertices in directed graph in a given systematic order. If a vertex has several neighbors,it would be equally correct to visit them in any order.By using the concepts of BFS,a heat transfer path is deﬁned that starts from one conﬂuence or division vertex,and trace the following verticesJ.Liu et al./International Journal of Refrigeration27(2004)965–973969according to the refrigerant ﬂow direction until the other conﬂuence or division one is met.When the refrigerant circuit splits into several branches from one division vertex,the model will search one of branch with the minimum start Tube No.until the other conﬂuence or division vertex is met,then it will go back to the previous division point to ﬁnish the remaining branch with the same method.Creation of heat transfer path is present as the following steps:Step 1:based on the adjacency matrix,the heat transfer path starts from the Tube No.0,and search for the next tube,until the conﬂuence or division vertex is met.Step 2:from the conﬂuence or division vertex,search for the next tube,until another conﬂuence or division vertex is met.Step 3:go on searching the vertices in the same way as step 2,until the last conﬂuence one is reached.As seen in the Fig.1,the heat transfer paths can be obtained as follows:0/8/44/3/2/54/7/6/55/1/93.1.2.Pressure drop pathThe DFS is another search algorithm,which considers the outgoing edges of a vertex before any neighbors of the vertex By using the concept of DFS,a pressure drop path is deﬁned that starts from one division vertex,and ends at the refrigerant outlet vertex.When the refrigerant circuit splits into several branches from one division vertex,the model will search one of branch with the minimum start Tube No.until the refrigerant outlet point is reached,then it will go back to the division point to ﬁnish the remaining branch with the same method.Creation of the pressure drop path is present as the following steps:Step 1:based on the adjacency matrix,the pressure drop path starts from the Tube No.0,and search the next tube until the refrigerant outlet point is reached.Step 2:from the division point,search for the next tube,until the outlet point is reached.Step 3:go on searching the vertices in the same way as step 2,until the last division vertex has ﬁnished.As seen in the Fig.1,the corresponding pressure drop path can be obtained as follows:0/8/4/3/2/5/1/94/3/2/5/1/94/7/6/5/1/93.2.Heat transfer calculation processProcess of heat transfer calculation starts from the inlet refrigerant tubes,and calculates all the control volumes in the heat exchange one by one along the heat transfer path until the last one is completed.By giving the inlet air state parameters of tubes in ﬁrst row,which are exposed to the upwind air,the inlet air state parameters of remaining tubes in other rows will be calculated depending on the outlet air state parameters from the upstream tubes.3.2.1.Heat transfer algorithm of control volumeFor a control volume,the simulation process begins with a set of inlet known parameters and assumed outlet unknowns on both air and refrigerant side.By assuming the outlet refrigerant speciﬁc enthalpy h r,out ,outlet air temperature T a,out ,and tube wall temperature T wall ,the energy governing equations can be solved.A two-level iteration method is developed to solve the set of equations ofTable 1List of applied heat transfer and pressure drop correlations Applied region ItemsCorrelations EvaporationCondensationRefrigerant sideHeat transfer coefﬁ-cientGabrielii and Vamling [7]for two phase,smooth tubeShah [8]for two phase,smooth tube Kandlikar [9]for two phase,enhanced tube Yu and Koyama [10,11]for two phase,enhanced tubeDittus–Boelter [8]for single phaseDittus–Boelter for single phasePressure dropJung (1989)[12]for two phase,smooth tubeM.Goto et al.(2001)[13]for two phase,smooth tubeCheng–Shu Kuo [14]for two phase,enhanced tubeModiﬁed Smith et al.(2001)[15]for two phase,enhanced tubeColebrook–White for single phase [15]Smith et al.(2001)[15]for single phaseAir sideHeat transfer coefﬁ-cientWang et al.(1999)[16]for wavy ﬁns Pressure dropWang et al.(1999)[16]for wavy ﬁnsJ.Liu et al./International Journal of Refrigeration 27(2004)965–973970a single control volume.Theﬁrst level iteration is to search a suitable temperature of tube wall.The second level iteration is to search suitable outlet air dry bulb temperature and outlet refrigerant speciﬁc enthalpy.Theﬂow chart is shown as Fig.5.3.2.2.Overall heat transfer calculationFor theﬁrst iteration loop,the heat exchange in each control volume can be calculated along heat transfer path one-by-one with refrigerant pressure in each control volume unchanged.After every control volume is computed,the inlet and outlet air parameters of each control volume are updated;the inlet and outlet refrigerant parameters (enthalpy and temperature)of each control volume are updated,which will be used by pressure drop calculation process.For subsequent iterations,the heat exchange of each control volume is calculated based on the adjusted refrigerantﬂow rate,which is determined by pressure drop calculation.In the process of iteration,the inlet air properties of each control volume are replaced by the upstream air state.3.3.Pressure drop calculation processProcess of pressure drop calculation starts from the inlet tubes,and calculates all the control volumes in the heat exchange along the pressure drop path one-by-one untilthe Fig.5.Flow chart of heat transfer calculation for one controlvolume.Fig.4.Logicalﬂow chart of a simulation algorithm.J.Liu et al./International Journal of Refrigeration27(2004)965–973971last one is completed.At the outset of the calculation,the pressure drop of each path is calculated one by one based on the inlet and outlet refrigerant states (enthalpy and temperature)of each control volume,which are calculated by the heat transfer calculation process.After every control volume is computed,the distribution of refrigerant mass ﬂow rate is adjusted to ensure the pressure drops of the pressure drop paths with same start point is the same,the inlet and outlet refrigerant pressure ofeach control volume are updated,which will be used by the heat transfer calculation process.4.Validation of the modelIn order to verify the reliability and accuracy of this model,the experimental test has been conducted on a actual heat exchanger,which is shown in Fig.6.This heat exchanger has 3rows and 16column tubes.These tubes are assembled with stagger type.The conﬁguration parameters are listed as Table 2.Experimental and predicted values are compared for a number of operating conditions.There are total 40test data used under various conditions operating as evaporator and condenser:refrigerant mass ﬂow rate is from 40to 180kg h K 1,air upwind velocity is from 0.5to 1.5m s K 1,the evaporation saturation temperature is from 4to 168C,and condensation saturation temperature is from 35to 608C.Fig.7shows the deviations between predicted heat exchanges and experimental values are within G 10%.Table 2Conﬁguration parameters for heat exchanger Geometric par-ameters Value Geometric par-ameters Value Geometric par-ametersValue Number of rows 3Fin corrugation angle158Micro ﬁn apex angle 608Transverse tube pitch25mm Fin collar outside diameter9.68mmMicro ﬁn space0.20mmLongitudinal tube pitch21.65mm Upwind area 0.229m 2Micro-ﬁn tube Smooth tube Outside diameter 9.40mm Outside diameter 9.00mm Wavy ﬁn geometric Tube wall thickness 0.26mm Tube wall thickness0.35mmFin thickness 0.14mm Micro ﬁn height 0.22mm Fin space1.86mmMicro ﬁn helix angle158Fig.6.The schematic diagram of refrigerantcircuitry.Fig.8shows the deviations between predicted pressure losses and experimental values for evaporator are within G 20%.It also shows that the pressure losses of condenser are very small and the absolute erros of the predicted pressure losses of condenser are less than30kPa.Such small error of pressure loss in condenser has almost no inﬂuence on the performance of heat exchangers.Figs.7and8show that the accuracy of the model is acceptable.As the accuracy of the model depends on the accuracies of correlations for heat transfer coefﬁcients and pressure losses,more accurate correlations are needed if better prediction accuracy is required.5.ConclusionsWith help of the concepts of directed graph and graph-based search algorithms(BFS and DFS)in graph theory,a general steady state distributed parameter model for theﬁn-and-tube heat exchanger is developed.The model makes it possible to analyze the performance of heat exchangers with complex refrigerant circuits.A novel computational algor-ithm,named alternative iteration method,is also developed to shorten the simulating time by decoupling the calculation of the energy conservation equations and momentum conservation equations.The predicted heat exchanges agree with experimental ones within a maximum error G 10.0%.The predicted refrigerant pressure drop for evapor-ator agree with experimental ones within a 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