Heat Transfer Exam 2018/2019 - University Ibn Tofail

# Université Ibn Tofail ## Ecole Nationale des Sciences Appliquées de Kénitra ### Filière Génie Mécatronique d'Automobile #### Examen de transfert thermiques ##### Année universitaire 2018/2019 ## Exercice The plane wall of a vehicle, exposed to the sun, receives a solar thermal power surfacic Ps=...

# Université Ibn Tofail ## Ecole Nationale des Sciences Appliquées de Kénitra ### Filière Génie Mécatronique d'Automobile #### Examen de transfert thermiques ##### Année universitaire 2018/2019 ## Exercice The plane wall of a vehicle, exposed to the sun, receives a solar thermal power surfacic Ps= 380W.m-2 assumed uniform and reflects a fraction a = 40%. A air conditioner maintains the interior air of the vehicle at the temperature T1 = 20°C while the exterior interior air is at the constant temperature TO = 35°C. The wall of the vehicle with thickness I=6cm is made of a homogeneous material with thermal conductivity λ = 0.18.m²¹.K-1; the heat exchanges between the air and each of the faces of the wall are of a conducto-convective nature and characterized by the coefficient h = 12 W.m-2.K-1. We will denote Ti and Te the temperatures of the wall surfaces in contact with the interior air and exterior air respectively. We will assume that all heat transfers occur along the Ox axis of the wall, and we study the stationary regime. 1. Show that the temperature gradient is constant inside the partition; express this gradient as a function of Ti, Te and I. 2. Reasoning on the heat flux surfacique, $1 - a$ h λ a- Express the sum Te+Ti as a function of To, T1, Ps and a = ; b- Express the temperatures Te and Ti as a function of the data Ps, To, T1, a and b = 1; c- Calculate numerically Te and Ti. 3. Establish the expression of the temperature T(x) inside the partition as a function of Te, Ti, I and the abscissa x (with x=0 on the wall-interior air interface). Plot T(x). 4. Calculate the power of the air conditioning for a wall surface S=10m² ## Problem A metallic sphere of center O and radius R is immersed in a liquid of mass volume µ, specific heat c and thermal conductivity λ. The sphere and the liquid are initially at the same temperature To uniform. At time t=0, the sphere is suddenly brought to the temperature T1 (T1>To) maintained constant at later times t>0. We will ignore the heat transfers inside the sphere, and study the response of the liquid medium to this « temperature step >>. 1. From the heat equation, write the differential equation satisfied by the temperature field T(r,t) at times t 20 in any point M (OM=r) of the liquid medium. R 2. We set T(r,t) = To +. (r,t). Establish the differential equation that r connects the partial derivatives ∂2θ ∂θ λ and of the thermal diffusivity K = of the liquid medium. ∂r2 ∂t µc 3. We associate to point M(OM=r), at time t, the positive spatio-temporal dimensionless parameter: u = Establish the second order differential equation r-R 2√Kt satisfied by the function (u). 4. Write the boundary conditions, in space and in time, of the function (u). 5. Determine the law (u), then the law of temperature distribution T(r,t) in the liquid using the error function: erf(u) = 2 Se-x² dx 6. a) What does the law of distribution T(r,t) become after a long enough time (t → ∞) ? b) What then is, if t→ ∞, of the expression of the heat flux through a sphere of center O and radius r>R, as a function of λ, R, To and T1. Remark: We give the Laplacian of a function f(r) in spherical coordinates: Af(r) = 1 d(r20f) r2 dr We are given erf(0)=0 ; erf(0.5)=0.52, erf(x)=1 and erf(u)= 1.1u for 0<u<0.4 . ![A diagram of a sphere submerged in a liquid] The sphere has a radius of R and its center is marked "O". The liquid is marked "µ, c ,λ". An arrow points upwards from the sphere toward the liquid and the liquid is also marked "T(r,t)".