When viscous dissipation is appreciable, the conservation of energy [Eq. 5.6] must take into account the rate at which mechanical energy is irreversibly converted to thermal energy due to viscous effects in the fluid. This gives rise to an additional term, Ø , on the right-hand side, the viscous dissipation where

Question

Apply the resulting equation to laminar flow between two infinite parallel plates, with

the upper plate moving at a velocity U. Assuming constant physical properties [p, cp, k,

?], obtain expressions for the velocity and temperature distributions. Compare the

solutions with the dissipation term included with the results when dissipation is

neglected. Find the plate velocity required to produce a 1 K temperature rise in

nominally 40°C air relative to the case where dissipation is neglected.



GIVEN

Laminar flow between two infinite parallel plates

Upper plate moves at a velocity U?

The viscous dissipation term given above must be used in the conservation of energy equation

FIND

(a) Expression for velocity and temperature distributions

(b) Compare these to solutions without the dissipation term

(c) Plate velocity that gives a 1 K rise in 40°C air relative to the case without the dissipation term

ASSUMPTIONS

Steady state

Constant physical properties

The plates are at constant temperatures, T1, T2

SKETCH



PROPERTIES AND CONSTANTS

From Appendix 2, Table 28, for dry air at 40°C

Thermal conductivity (k) = 0.0265 W/(m K)

Absolute viscosity (?) = 19.1 x 10–6 N s/m2

Answer 1

(a) Including the viscous dissipation term in Equation (5.6)



Eliminating the terms which are zero for this case



(Note that since the left side of Equation (5.6) drops out completely, d

2T/dy2 is multiplied by k and not by a For this case, the conservation of momentum Equation (5.5) reduces to:



The boundary conditions for these equations are



Integrating the momentum equation twice yields



Applying the first boundary condition: c2 = 0

Applying the second boundary condition: c1 = U/H

Therefore, the velocity distribution between the plates is



Substituting this into the energy equation yields



Integrating twice



Applying the first boundary condition: c2 = T1

Applying the second boundary condition: c1 = (T2 – T1)/H + (U)/(2 k H)

Therefore, the temperature distribution is



(b) When dissipation is neglected, the momentum equation, and therefore, the velocity distribution, remain unchanged. Without viscous dissipation, the energy equation



Integrating twice



From the first boundary condition: c2 = T1

From the second boundary condition: c1 = (T2 – T1)/H



Including the viscous dissipation term leads to an increase in temperature



at each distance y from the lower plate.

(c) This temperature increase is a maximum at



At this point, the temperature increase is



For ?T = 1 K