Convective heat transfer

At y = 0 It is said that the heat is transferred by conduction because the fluid has a velocity equal to 0 (no-slip conditions). But the temperature of the fluid for y=0 is equal to Ts(temperature of the surface) so how come there's a gradient in the temperature? Do they mean there's gradient at y=0 just above the layer of the fluid having a temperature equal to Ts? Another question, the conduction heat transfer at y = 0 is happening for 1s (J/S = W) but during this second the fluid is passing and it is not static like in a wall that issue is confusing me. Even the exchange of energy between the fluid that is not on the surface, this fluid is flowing and heat is transferred each second by Qs = h(Ts-Tinf) and on each position x we have a different gradient of temperature that is independent of the time but this variation by position is due to heat transfer. I am very confused about the steady state heat transfer by convection.

Hope someone can clarify it

At the wall surface $T=T_s$. We know that across an infinitesimally thin layer $dy$ temperature drops by $dT$, so by definition the temperature gradient at $y=0$ is:

$\large{\frac{\partial T}{\partial y}_{y=0}}$

From this we can easily derive:

$\Large{h=\frac{-k_f\frac{\partial T}{\partial y}_{y=0}}{T_s-T_{\infty}}}$

But note that this doesn't tell the full story: $h$ is only the heat transfer coefficient at the boundary and would be used in approximate calculations like this one. In that derivation the temperature gradient was not taken into account and the temperature of the body considered homogeneous ($\frac{\partial T}{\partial y}=0$). But for a detailed calculation including temperature gradient across the body, Fourier's Law has to be applied throughout the rest of the body too, not just the boundary layer.

• Great. But what I am not understanding that the thermal boundry layer change with position x. I am not understanding the fact for a time equal one second ( because of heat flux has a unit equal to 1 J/S) and for a position x, the fluid will flow so we dont have a static object like in a wall. For different position of x we have different temperature. Why we have this difference of temperature and how it can be change independatly of the time because we are in a steady stat – Tonylb1 Oct 13 '15 at 16:51
• @Tonylb1: Hi. Let's try this step-by-step (Q&A). Do you understand that the boundary layer is stationary? This is a basic premise of all contained flow (laminar or tubulent) that at $y=0$, $\frac{dv}{dy}=0$ and $v=0$ (with $v$ flow speed in $\text{m/s}$). – Gert Oct 13 '15 at 17:03
• Ok but above this layer, the fluid is flowing and on each x we have different temperature. Hest flux is by watt by second but during this second the fluid will flow that's why it is confusing me about steady stat and temperature variation with position x – Tonylb1 Oct 13 '15 at 18:03
• @Tonylb1: the term steady state may appear a little misleading here. Imagine water flows at constant rate through a circular pipe at constant $T_s$. Things inside the pipe are decidedly dynamic: a flow speed gradient $\frac{\partial v}{dx}$ is established, as well as temperature gradients $\frac{\partial T}{\partial y}$ and $\frac{\partial T}{\partial x}$. Also a net heat flow $q$ flows from the pipe to the water. But if you keep all parameters constant then also all gradients and $q$ will remain unchanged. That's why we call it steady state. – Gert Oct 13 '15 at 18:27
• so in this second "new particles" of fluid will replace an old one with same temperature right? – Tonylb1 Oct 13 '15 at 18:41