7. The finite-difference formulation of a typical interior node in for steady, 2-D heat conductionwith a constant thermal conductivity in rectangular coordinates is expressed as follows:'m-1,n – 2Tm,n + Tm+1,n , Tm,n-1– 27m,n + Tm,n+:т,п+1(Ax)2(Ay)?a. Modify the above difference equation for a medium with heat generation. Show all thedetails before arriving at the final solution.b. Modify the difference equation given in part (a) for the 3-D case, but don't include heatgeneration.

7. The finite-difference formulation of a typical interior node in for steady, 2-D heat conduction with a constant thermal conductivity in rectangular coordinates is expressed as follows: Tm-1,n – 2Tm,n + Tm+1,n , Tm,n-1– 27m,n + Tm,n+1 т,п (Ax)² (Ay)? a. Modify the above difference equation for a medium with heat generation. Show all the details before arriving at the final solution. b. Modify the difference equation given in part (a) for the 3-D case, but don't include heat generation.

7. The finite-difference formulation of a typical interior node in for steady, 2-D heat conduction with a constant thermal conductivity in rectangular coordinates is expressed as follows: Tm-1,n – 2Tm,n + Tm+1,n , Tm,n-1– 27m,n + Tm,n+1 т,п (Ax)² (Ay)? a. Modify the above difference equation for a medium with heat generation. Show all the details before arriving at the final solution. b. Modify the difference equation given in part (a) for the 3-D case, but don't include heat generation.

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter4: Numerical Analysis Of Heat Conduction

Section: Chapter Questions

Problem 4.57P

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2.29 In a cylindrical fuel rod of a nuclear reactor, heat is generated internally according to the equation where = local rate of heat generation per unit volume at r = outside radius = rate of heat generation per unit volume at the centerline Calculate the temperature drop from the centerline to the surface for a 2.5-cm-diameter rod having a thermal conductivity of if the rate of heat removal from its surface is 1.6 .
3.16 A large, 2.54-cm.-thick copper plate is placed between two air streams. The heat transfer coefficient on one side is and on the other side is . If the temperature of both streams is suddenly changed from 38°C to 93°C, determine how long it takes for the copper plate to reach a temperature of 82°C.
2.9 In a large chemical factory, hot gases at 2273 K are cooled by a liquid at 373 K with gas-side and liquid-side convection heat transfer coefficients of 50 and , respectively. The wall that separates the gas and liquid streams is composed of a 2-cm thick oxide layer on the gas side and a 4-cm thick slab of stainless steel on the liquid side. There is a contact resistance between the oxide layer and the steel of . Determine the rate of heat loss from hot gases through the composite wall to the liquid.
2.38 The addition of aluminum fins has been suggested to increase the rate of heat dissipation from one side of an electronic device 1 m wide and 1 m tall. The fins are to be rectangular in cross section, 2.5 cm long and 0.25 cm thick, as shown in the figure. There are to be 100 fins per meter. The convection heat transfer coefficient, both for the wall and the fins, is estimated to be K. With this information determine the percent increase in the rate of heat transfer of the finned wall compared to the bare wall.
Show that the rate of heat conduction per unit length through a long, hollow cylinder of inner radius ri and outer radius ro, made of a material whose thermal conductivity varies linearly with temperature, is given by qkL=TiTo(rori)/kmA where Ti = temperature at the inner surface To = temperature at the outer surface A=2(rori)/ln(ro/ri)km=ko[1+k(Ti+To)/2]L=lenthofcyclinder
1.10 A heat flux meter at the outer (cold) wall of a concrete building indicates that the heat loss through a wall of 10-cm thickness is . If a thermocouple at the inner surface of the wall indicates a temperature of 22°C while another at the outer surface shows 6°C, calculate the thermal conductivity of the concrete and compare your result with the value in Appendix 2, Table 11.
A steam pipe 200 mm in diameter passes through a large basement room. The temperature of the pipe wall is 500C, while that of the ambient air in the room is 20C. Determine the heat transfer rate by convection and radiation per unit length of steam pipe if the emissivity of the pipe surface is 0.8 and the natural convection heat transfer coefficient has been determined to be 10 W/m2K.
An electronic device that internally generates 600 mW of heat has a maximum permissible operating temperature of 70C. It is to be cooled in 25C air by attaching aluminum fins with a total surface area of 12cm2. The convection heat transfer coefficient between the fins and the air is 20W/m2K. Estimate the operating temperature when the fins are attached in such a way that (a) there is a contact resistance of approximately 50 K/W between the surface of the device and the fin array and (b) there is no contact resistance (in this case, the construction of the device is more expensive). Comment on the design options.
The handle of a ladle used for pouring molten lead is 30 cm long. Originally the handle was made of 1.9cm1.25cm mild steel bar stock. To reduce the grip temperature, it is proposed to form the handle of tubing 0.15 cm thick to the same rectangular shape. If the average heat transfer coefficient over the handle surface is 14 W/m K, estimate the reduction of the temperature at the grip in air at 21C.
EXAMPLE 1| Aspherical container of inner radius r1 =2 m, outer radius 2= 2.1 m, ‘and thermal conductivity k=30 Wim - °C is illed with iced water at 0°C e container is gaining heat by convection from the surrounding air at 7 = 25°C with a heat transfer coeflicient of h = 18 W/m2 - °C. Assuming the inner surface temperature of the container to be 0°C, ) express the differential equation and the houndary conditions for steady one-dimensional heat conduction through the container ) obtain a relation for the variation of temperature in the cantainer by solving the differential equation €) evaluate the rate of heat gain 10 the iced water.
3. A food product with 80% moisture content in a 7 cm diameter can wants to be frozen. The density of the product is 1000 kg / m³, the thermal conductivity is 1.0 W / (m K), and the initial freezing temperature is -2 ° C. After 11 hours in the freezing medium -30 ° C, the product temperature becomes -10 ° C. Estimate the convection heat transfer coefficient of the freezing medium. Assume the can as an infinite cylinder. h = Answer W / (m² K).
Water at 5°C is being used to cool apples from an initial temperature of 25° to 10°C. Thewater flow over the surface of the apple creates a convective heat-transfer coefficientof 15 W/(m2-K). Assume the apple can be described as a sphere with a 10 cm diameterand the geometric center is to be reduced to 10°C. The apple properties include thermalconductivity of 0.45 W/(m-K), specific heat of 3.950 kJ/(kg-K), and density of 940 kg/m3.a. Determine the time that the apples must be exposed to the water.b. If the apple is submerged only for 2 hours, what will be its center temperature?