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Derive general heat conduction equation in Cartesian coordinates? Consider a small rectangular element of sides dx, dy and dz as shown in figure. No heat generation and steady state: General heat conduction equation in Cylindrical coordinates: While dealing with problems of conduction of heat through systems having cylindrical geometrics it is convenient to use cylindrical coordinates. Consider an elemental volume having the coordinates r, , z for three dimensional heat conduction analysis as shown in figure. Net heat accumulated in the element due to conduction of heat from all the coordinate direction considered: Heat flow in radial direction r plane: Heat influx dr krd Qr r T dz. T r dz kdrrd Heat flow in axial direction z plane: Heat influx:

Moisture content. Density of material. Pressure and temperature. What is super insulation and give its application.

Super insulation is a process which is used to keep the cryogenic liquids at very low temperature. The super insulation consists of multiple layers of highly reflective material separated by insulating spacers. The entire system is evacuated to minimize air conduction. Where do you use Heislers chart. Heislers charts are used to solve problems Transient heat conduction in solids with finite conduction and convective resistances.

Give some examples of heat generation application in heat conduction. Fuel rod nuclear reactor. Electrical conductor. Chemical and combustion process. Drying and setting of concrete. State Newtons law of cooling or convection law. Write down the equation for heat transfer through a composite plane wall. R T Q Transfer Heat overall. Write down the equation for heat transfer through composite pipes or cylinder.

Write down one dimensional, steady state conduction equation without internal heat generation. Write down steady state, two dimensional conduction equation without heat generation. Write down the general equation for one dimensional steady state heat transfer in slab or plane wall without heat generation t T z T y T x T. Define overall heat transfer co-efficient. The overall heat transfer is defined as amount of transmitted per unit area per unit time per degree temperature difference between the bulk fluids on each side of the metal.

Define fins or Extended surfaces. It is possible to increase the heat transfer rate by increasing the surface of heat transfer. The surfaces used for increasing heat transfer are called extended surfaces or sometimes known as fins. State the applications of fins. The main applications of fins are 1. Cooling of electronic components 2. Cooling of motor cycle engines. Cooling of transformers.

Cooling of small capacity compressors. Define Fin efficiency. Define Fin effectiveness. Fin effectiveness is the ratio of heat transfer with fin to that without fin without fin with Q Q ess effectiven Fin What is meant by steady stale heat conduction?

If the temperature of a body does not vary with time, it is said to be in a steady state and that type of conduction is known as steady state heat conduction.

What is meant by Transient heat conduction or unsteady state conduction? If the temperature of a body varies with time, it is said to be in a transient state and that type of conduction is known as transient heat conduction or unsteady state conduction. What is Periodic heat flow? In periodic heat flow, the temperature varies on a regular basis. Cylinder of an IC engine. Surface of earth during a period of 24 hours. What is non periodic heat flow?

In non periodic heat flow, the temperature at any point within the system varies non linearly with time. Heating of an ingot in a furnace. Cooling of bars. What is meant by Newtonian heating or cooling process?

The process in which the internal resistance is assumed as negligible in comparison with its surface resistance is known as Newtonian heating or cooling process. What is meant by Lumped heat analysis? In a Newtonian heating or cooling process the temperature throughout the solid is considered to be uniform at a given time.

Such an analysis is called Lumped heat capacity analysis. What is meant by Semi-infinite solids? In a semi infinite solid, at any instant of time, there is always a point where the effect of heating or cooling at one of its boundaries is not felt at all. At this point the temperature remains unchanged. In semi infinite solids, the biot number value is.

What is meant by infinite solid? A solid which extends itself infinitely in all directions of space is known as infinite solid. In infinite solids, the biot number value is in between 0. Define Biot number. It is defined as the ratio of internal conductive resistance to the surface convective resistance. What is the significance of Biot Number? Explain the significance of Fourier number. It is defined as the ratio of characteristic body dimension to temperature wave penetration depth in time.

It signifies the degree of penetration of heating or cooling effect of a solid. What are the factors affecting the thermal conductivity? Structure of material. Explain the significance of thermal diffusivity.

The physical significance of thermal diffusivity is that it tells us how fast heat is propagated or it diffuses through a material during changes of temperature with time. If the temperatures of the inner and outer surface are maintained at 50C and 30C respectively. Calculate the heat loss through one square meter area. Find also the temperature at an interior point of the wall 24cm distance from the outer wall. A steam pipe mm OD is covered with 25mm thich layer of insulation material with an average thermal conductivity of 0.

The temperature of the pipe surface is C and that of the outer surface of insulation is 50C. Find the loss of heat from a length of 10m of the pipe line. Compute the heat loss per square meter of the surface area of furnace wall 25cm thick. The inner and outer surface temperatures are C and 40C respectively. Find 1 The rate of heat flow at the inside and outside surface per m length.

An aluminium sphere weighing 7kg and initially at a temperature of C is suddenly immersed in a fluid at 10C. The energy transfer in convection is predominately due to the bulk motion of the fluid particles; through the molecular conduction within the fluid itself also contributes to some extent.

If this motion is mainly due to the density variations associated with temperature gradients within the fluid, the mode of heat transfer is said to be due to free or natural convection. On the other hand if this fluid motion is principally produced by some superimposed velocity field like fan or blower, the energy transport is said to be due to forced convection. Velocity Boundary Layer: Consider the flow of fluid over a flat plate as shown in the figure.

The fluid approaches the plate in x direction with uniform velocity u. The fluid particles in the fluid layer adjacent to the surface get zero velocity. This motionless layer acts to retart the motion of particles in the adjoining fluid layer as a result of friction between the particles of these two adjoining fluid layers at two different velocities. The region of the flow over the surface bounded by in which the effects of viscous shearing forces caused by fluid viscosity are observed, is called velocity boundary layer or hydro dynamic boundary layer.

The retardation of fluid motion in the boundary layer is due to the shear stresses acting in opposite direction with increasing the distance y from the surface shear stress decreases, the local velocity u increases until approaches u. With increasing the distance from the leading edge, the effect of viscosity penetrates further into the free stream and boundary layer thickness grows. Thermal boundary Layer: If the fluid flowing on a surface has a different temperature than the surface, the thermal boundary layer developed is similar to the velocity boundary layer.

Consider a fluid at a temperature T. The fluid particles in adjacent layer to the plate get the same temperature that of surface. The particles exchange heat energy with particles in adjoining fluid layers and so on. As a result, the temperature gradients are developed in the fluid layers and a temperature profile is developed in the fluid flow, which ranges from Ts at the surface to fluid temperature T sufficiently far from the surface in y direction. The thickness of thermal boundary layer th at any location along the length of flow is defined as a distance y from the surface at which the temperature difference T-Ts equal 0.

With increasing the distance from leading edge the effect of heat transfer penetrates further into the free stream and the thermal boundary layer grows as shown in the figure. The convection heat transfer rate any where along the surface is directly related to the temperature gradient at that location. Therefore, the shape of the temperature profile in the thermal boundary layer leads to the local convection heat transfer between surface and flowing fluid.

Development of velocity boundary layer on a flat plate: It is most essential to distinguish between laminar and turbulent boundary layers.

Initially, the boundary layer development is laminar as shown in figure for the flow over a flat plate. Depending upon the flow field and fluid properties, at some critical distance from the leading edge small disturbances in the flow begin to get amplified, a transition process takes place and the flow becomes turbulent. In laminar boundary layer, the fluid motion is highly ordered whereas the motion in the turbulent boundary layer is highly irregular with the fluid moving to and from in all directions.

Due to fluid mixing resulting from these macroscopic motions, the turbulent boundary layer is thicker and the velocity profile in turbulent boundary layer is flatter than that in laminar flow. Depending upon surface and turbulence level of free stream the critical Reynolds number varies between 10 5 and 3 X 10 6. In the turbulent boundary layer, as seen three distinct regimes exist. A laminar sub- layer, existing next to the wall, has a nearly linear velocity profile.

The convective transport in this layer is mainly molecular. In the buffer layer adjacent to the sub-layer, the turbulent mixing and diffusion effects are comparable. Then there is the turbulent core with large scale turbulence. What is dimensional analysis? Dimensional analysis is a mathematical method which makes use of the study of the dimensions for solving several engineering problems. This method can be applied to all types of fluid resistances, heat flow problems in fluid mechanics and thermodynamics.

State Buckingham theorem. Buckingham theorem states as follows: These dimensionless terms are called terms. What are all the advantages of dimensional analysis? It expresses the functional relationship between the variables in dimensional terms.

It enables getting up a theoretical solution in a simplified dimensionless form. The results of one series of tests can be applied to a large number of other similar problems with the help of dimensional analysis. What are all the limitations of dimensiona1 analysis? The complete information is not provided by dimensional analysis.

It only indicates that there is some relationship between the parameters. No information is given about the internal mechanism of physical phenomenon. Dimensional analysis does not give any clue regarding the selection of variables. Define Reynolds number Re. It is defined as the ratio of inertia force to viscous force.

Define Prandtl number Pr. It is the ratio of the momentum diffusivity to the thermal diffusivity.

Define Nusselt Number Nu. It is defined as the ratio of the heat flow by convection process under a unit temperature gradient to the heat flow rate by conduction under a unit temperature gradient through a stationary thickness L of metre. Define Grashof number Gr. It is defined as the ratio of product of inertia force and buoyancy force to the square of viscous force. Define Stanton number St. It is the ratio of Nusselt number to the product of Reynolds number and Prandtl number.

Pr Re. Nu St What is meant by Newtonian and non-Newtonian fluids? The fluids which obey the Newton's law of viscosity are called Newtonian fluids and those which do not obey are called non Newtonian fluids.

What is meant by laminar flow and turbulent flow? Laminar flow is sometimes called stream line flow. In this type of flow, the fluid moves in layers and each fluid particle follows a smooth continuous path. The fluid particles in each layer remain in an orderly sequence without mixing with each other. Turbulent flow: In addition to the laminar type of flow, a distinct irregular flow is frequently observed in nature. This type of flow is called turbulent flow.

The path of any individual particle is zigzag and irregular. What is hydrodynamic boundary layer? What is thermal boundary layer?

Define convection. State Newtons law of convection. What is meant by free or natural convection? If the fluid motion is produced due to change in density resulting from temperature gradients, the mode of heat transfer is said to be free or natural convection. What is forced convection? If the fluid motion is artificially created by means of an external force like a blower or fan, that type of heat transfer is known as forced convection. According to Newton's law of cooling the amount of heat transfer from a solid surface of area A, at a temperature T, to a fluid at a temperature T, is given by T T hA Q W What are the dimensionless parameters used in forced convection?

Reynolds number Re. Nusselt number Nu.

Prandtl number Pr. Define boundary layer thickness. Indicate the concept or significance of boundary layer. In the boundary layer concept the flow field over a body is divided into two regions: A thin region near the body called the boundary layer where the velocity and the temperature gradients are large.

The region outside the boundary layer where the velocity and the temperature gradients are very nearly equal to their free stream values. Sketch the boundary development of a flow. Define displacement thickness.

Define momentum thickness. The momentum thickness is defined as the distance through which the total loss of momentum per second is equal to if it were passing a stationary plate.

Define energy thickness. The energy thickness can be defined as the distance, measured perpendicular to the boundary of the solid body, by which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid on account of boundary layer formation.

The base temperature of the rod is maintained at C. Estimate the heat transfer rate from the rod to the surrounding air. Also compare the relative heat flows and fin efficiencies with respect to the copper fin. A Copper wire 0. Determine the temperature at the depth of Also calculate the energy removed per unit area from the plate during 1 min of immersion.

Air flows over a thin plate with a velocity of 2. The width of the plate is 1m and its length is also 1m. The plate is 75cm long and is maintained at 60C. Also calculate the turbulent boundary layer thickness at the end of the plate assuming it to develop from the leading edge of the plate. Air at a temperature of Calculate the convective heat transfer coefficient. It produces 40W. Estimate the heat transfer coefficient and compute the percentage of power lost due to convection.

In a pressurized water space heater, heated water is passed through a staggered tube arrangement for which the tube outside diameter is There are 7 rows of tubes in the air flow direction. An instant water heater consists of a 4 mm I. Tube through which water flows at the rate of 3. A nichrome heating element wound over the tube provides a constant heat flux of W per metre length into the water.

Find the length of the tube to raise the temperature to 75C and also the maximum temperature at the exit. Ethylene glycol enters a 5 m length of mm diameter copper tube in a cooling system at a velocity of 5 mls. Estimate the heat transfer rate if the average bulk temperature is 20C and the tube wall is maintained at C.

Hot air at C flows through a duct of 15 cm diameter with a mass flow rate of 0. The temperature of air at a distance of 5 m from entry has been measured to be 77C. Neglecting the duct wall resistance calculate the heat loss from the duct over the 5 m length. In a double pipe-heat exchanger steam flows through the inner pipe and the air through the annular space.

The outer diameter of the inner pipe is 25 cm and the inner diameter of the outer pipe which is insulated is 38 cm. If the steam is condensing at C on the inner surface of the inner tube, estimate the heat transfer coefficient on the air side. A vertical plate 0. Compute the boundary layer thickness and the average heat transfer coefficient at the trailing edge of the plate. An un-insulated duct of width 0. If the surface temperature of the duct is maintained at 45C, compute the loss of heat from the duct per meter length.

A cylindrical heating element The surface of the heating element is maintained at a uniform temperature of Calculate the mean heat transfer coefficient and the rate of heat loss by the free convection from the entire surface of the element to the water. A steam pipe of mm outer diameter is placed horizontally in a room at 23C. The outside surface temperature of the pipe is C and its emissivity is 0. Determine the rate of heat loss per unit length of the pipe. The temperature of the tube is maintained at 90C and the length of the tube is cm.

Calculate the heat transfer coefficient using Hausen'scorrelation. Also find the mean temperature of cheese leaving the heated section. Find the heat transfer coefficient and the length of the tube required to meet the above requirement of heat.

The resistances of tube and film may be neglected. Water flows in a 50 mm diameter tube 3 m long at an average temperature ,of 30C. The tube wall temperature is maintained at 70C and the flow velocity is 0. Estimate the heat transfer coefficient using the Dittus-Boelter correlation, A horizontal steam pipe of 0.

The outside surface temperature is 80C and the emissivity of the pipe material is 0. Estimate the total heat loss from the pipe per metre length due to free convection and radiation. One surface of a panel 2m X 2m is insulated and the other surface is kept at a uniform temperature of 95C. Calculate the mean heat transfer coefficient due to free convection between the heated surface of the panel and the atmospheric air at 10C when Heated surface is vertical Panel is horizontal with hot surface facing up Panel is horizontal with hot surface facing down.

Boiling may occur when a liquid is in contact with a surface maintained at a temperature higher than the saturation temperature of the liquid. If heat is added to a liquid from a submerged solid surface, the boiling process is referred to as pool boiling.

In this process the vapor produced may form bubbles, which grow and subsequently detach themselves from the surface, rising to the free surface due to buoyancy effects. A common example of pool boiling is the boiling of water in a vessel on a stove. In contrast, flow boiling or forced convection boiling occurs in a flowing stream and the boiling surface may itself be apportion of the flow passage.

This phenomenon is generally associated with two phase flows through confined passages. A necessary condition for the occurrence of pool boiling is that the temperature of the heating surface exceeds the saturation temperature of the liquid.

The type of boiling is determined by the temperature of the liquid. If the temperature of the liquid is below the saturation temperature, the process is called sub cooled or local boiling.

In local boiling, the bubbles formed at the surface eventually condense in the liquid. If the liquid is maintained at saturation temperature, the process is called saturated or bulk boiling. There are various distinct regimes of pool boiling in which the heat transfer mechanism differs radically.

The temperature distribution in saturated pool boiling with a liquid vapor interface is shown in the Figure1. The different regimes of boiling are indicated in Figure 2. This specific curve has been obtained from an electrically heated platinum wire submerged in water b y varying its surface temperature and measuring the surface heat flux q s. The six regimes of Figure 2 will now be described briefly. In region I, called the free convection zone, the excess temperature, T is very small and 5C.

Here the liquid near the surface is superheated slightly, the convection currents circulate the liquid and evaporation takes place at the liquid surface. As the excess temperature, T is increased, bubbles begin to form on the surface of the wire at certain localized spots. The bubbles condense in the liquid without reaching the liquid surface. Region II is in fact the beginning of nucleate boiling. As the excess temperating is further increased bubbles are formed more rapidly and rise to the surface of the liquid resulting in rapid evaporation.

This is indicated in region III. Nucleate boiling exists up to T 50 C. The trend of increase of heat flux with increase in excess temperature observed up to region III is reversed in region IV, called the film boiling region.

This is due to the fact that bubbles now form so rapidly that they blanket the heating surface with a vapor film preventing the inflow of fresh liquid from taking their place. Now the heat must be transferred through this vapor film by conduction to the liquid to effect any further boiling. Since the thermal conductivity of the vapor film is much less than that of the liquid, the value of q. In region IV the vapor film is not stable and collapses and reforms rapidly.

With further increase in T the vapor film is stabilized and the heating surface is completely covered by a vapor blanket and the heat flux is the lowest as shown in region V.

The surface temperatures required to maintain a stable film are high and under these conditions a sizeable amount of heat is lost by the surface due to radiation, as indicated in region VI. The phenomenon of stable film boiling can be observed when a drop of water falls on a red hot stove.

This is due to the formation of a stable steam film at the interface between the hot surface and the liquid droplet. From Fig. The equipment used for boiling should be designed to operate in this region only. The critical heat flux point A in Fig.

The temperature at point B is extremely high and normally above the melting point of the solid. So if the heating of the metallic surface is not limited to point A, the metal may be damaged or it may even melt. That is why the peak heat flux point is called the burnout point and an accurate knowledge of this point is very important. Our aim should be to operate the equipment close to this value but never beyond it.

Flow Boiling: Flow or forced convection boiling may occur when a liquid is forced through a channel or over a surface which is maintained at a temperature higher than the saturation temperature of the liquid. There are numerous applications of flow boiling in the design of steam generators for nuclear power plants and space power plants. The mechanism and hydronamics of flow boiling are much more complex than in pool boiling because the bubble growth and separation are strongly affected by the flow velocity.

The flow is a two-phase mixture of the liquid and its vapor. Heat transfer to the sub cooled liquid at entry is by forced convection. This regime continues until boiling starts. The heat transfer coefficient in the boiling regime is suddenly increased.

In this boiling regime, the bubbles appear on the heated surface, grow and are carried into the mainstream of the liquid, so that a bubbly flow regime prevails for some length of the tube. As the volume fraction of the vapor increases, the individual bubbles coalesce and plugs or slugs of vapor are formed.

This regime is called the slug flow regime. As the vapor quality is increased, the flow becomes annular with a thin liquid layer on the wall and a vapor core. The vapor velocity is much higher than that of the liquid.

The phenomenon of stable film boiling can be observed when a drop of water falls on a red hot stove. This is due to the formation of a stable steam film at the interface between the hot surface and the liquid droplet.

From Fig. The equipment used for boiling should be designed to operate in this region only. The critical heat flux point A in Fig. The temperature at point B is extremely high and normally above the melting point of the solid.

So if the heating of the metallic surface is not limited to point A, the metal may be damaged or it may even melt. That is why the peak heat flux point is called the burnout point and an accurate knowledge of this point is very important. Our aim should be to operate the equipment close to this value but never beyond it.

Flow Boiling: Flow or forced convection boiling may occur when a liquid is forced through a channel or over a surface which is maintained at a temperature higher than the saturation temperature of the liquid. There are numerous applications of flow boiling in the design of steam generators for nuclear power plants and space power plants.

The mechanism and hydronamics of flow boiling are much more complex than in pool boiling because the bubble growth and separation are strongly affected by the flow velocity. The flow is a two-phase mixture of the liquid and its vapor. Heat transfer to the sub cooled liquid at entry is by forced convection.

This regime continues until boiling starts. The heat transfer coefficient in the boiling regime is suddenly increased. In this boiling regime, the bubbles appear on the heated surface, grow and are carried into the mainstream of the liquid, so that a bubbly flow regime prevails for some length of the tube. As the volume fraction of the vapor increases, the individual bubbles coalesce and plugs or slugs of vapor are formed. This regime is called the slug flow regime.

As the vapor quality is increased, the flow becomes annular with a thin liquid layer on the wall and a vapor core.