Тепломассообмен 5
Есть решение задач 4 и 7, контакты
List of attestation tasks on “Heat and Mass Transfer”
1. Semi-infinite rod (x ≥ 0) is thermally insulated on its side surface and has a constant initial temperature of t0. At the initial time, the end of the rod is immersed in a medium with a constant temperature of tf. The heat transfer coefficient from the rod end to the fluid is constant and equals to α. The thermal diffusivity of the rod is a = const. Find the unsteady temperature distribution within the rod.
Recommendation: solve the problem using the Laplace transform method. To determine the inverse Laplace transform use the following formula:
where erf(x) is the Gauss error function defined as:
erf(x) = 1 – erf(x)
and erfc(x) is complementary error function which correlates to erf(x) as follows:
erfc(x) = 1 – erf(x)
2. Determine the unsteady temperature distribution t(x, τ) within a flat plate with a thickness of 2δ. The initial temperature of the plate is t0. At the initial time, the plate is immersed into a fluid with a constant temperature tf, which is not equal to t0. Heat transfer coefficients on the surfaces of the plate x = — δ and x = δ are the same and equal to α.
3. The body with a volume of V and an outer surface of F has an initial temperature of t0. At the initial time, this body is immersed into the fluid whose temperature increases linearly with time:
Tf(τ) = tf0 + kτ,
where k is the rate of change in fluid temperature. The initial temperature of the body t0 is greater than the initial temperature of fluid tf0, t0 > tf0. The heat transfer coefficient between the body surface and fluid is constant, and equal to α. All the thermophysical properties of the body (specific heat capacity, cp; density, ρ; thermal conductivity, λ; and thermal diffusivity, a) are assumed known and constant.
4. The steel sheet with a thickness of δ = 4 mm has an initial temperature t0 = 20°C and is cooled by a flow of gas at a temperature of tf = 20°C. Determine the temperature of the sheet after 5, 10, and 15 minutes have passed since the start of the cooling. The heat transfer coefficient from the sheet surface to the gas flow is α = 20W/(m2 · K). The thermophysical properties of the steel are as follows: density, ρ = 7900 kg/m3; specific heat capacity, cp = 462 J/(kg · K); thermal diffusivity, a = 1,25 · 10-5 m2/s.
5. Using the finite thickness boundary layer method, derive an expression for local and average heat transfer coefficients on the surface of a plate, which is longitudinally washed by a fluid. The surface has a constant temperature tp = const. Assume that the boundary layer on the plate is laminar and the physical properties of the liquid do not dependent on temperature, and the Prandtl number of the fluid is greater than 1. Compare the result to the exact solution:
Nux = 0.332 Rex0.5 Pr1/3
6. Consider the longitudinal flow of fluid along the flat plate, on the surface of which a constant temperature tw = const is maintained. The flow is laminar, and the fluid has a Prandtl number much less than 1 (Pr << 1). Derive an expression for local and average heat transfer coefficients on the surface of the plate neglecting the change in velocity within the thermal boundary layer and using the Laplace transform method to integrate the differential equation of energy transfer.
Recommendation: to determine the inverse Laplace transform use the following formula:
7. Determine an average heat transfer coefficient and heat flow on a flat roof with a length of 3 m and width of 1 m, which is blown by a longitudinal flow of air with a velocity of w0 = 1,5 m/s. The temperature of the roof is constant and equal to tw = 40°C. The ,temperature of incoming air flow is t0 = 20°C. The thermophysical properties of air required for solving the problem are as follows: thermal conductivity, λ = 0,0266 W/(m · K); kinematic viscosity, ν = 1,605 · 10-5 m2/s; Prandtl number, Pr = 0,7067.
Есть решение задач 4 и 7, контакты