Parallel Plate Heat Exchanger

Parallel Plate Heat Exchanger Parallel Plate Heat Exchangers are use in a number of thermal processing applications. The characteristics are that the fluids flow in the narrow gap, between two parallel sheets. The flow is usually laminar. For our example, assume a fluid flows confined between two parallel planes. both held at a fixed temperature, T H. The fluid enters at T 1 into the heated section at a mean velocity U. The flow is laminar so that the velocity profile is given by u ( y ) = 3 2 U 1 y 2 H The volumetric flow rate per unit width is given by q w = 2UH Reasonable assumptions include, Steady State, no viscous dissipation, constant thermal properties, etc. ChE 333 1

Uniform wall temperature T y = H H Fluid in laminar flow y x Uniform inlet temperature T 1 x= y = H Temperature profile T(x,y) Figure 12.5.1 Laminar flow between heated parallel planes The convective energy balance is given by u( y) T x = α 2 T y 2 + 2 T x 2 As we saw in the case of a tubular channel, radial conduction is much larger than axial conduction so the equation can be simplified to Boundary Conditions u(y) T x = α 2 T Dy 2 T = T1 at x = for all y Initial temperature profile T y = at y = for all x Symmetry at the channel axis T = TH at y H for all x Wall temperature ChE 333 2

Non-Dimensional Form of the Equation We can make the equation dimensionless with these definitions Θ = T T H T 1 T H ; y* = y H ; x*= αx UH 2 So that the equation and its boundary conditions become 3 2 1 y* 2 Θ = 2 Θ x* y* 2 Θ = 1 at x* = for all y* Θ y* = at y* = for all x* Θ= at y* = ±1 for all x* The solution of the equation can be expressed as a series solution Θ = Σ m= A n exp 2 3 λ 2 nx* a nm y* n Σn= The heating rate per unit width of channel is Q H = 2 U H [ρ Cp (T 1 T cm )] where T cm is the mixing cup average temperature. ChE 333 3

The mixing cup average obtained from the temperature profile is Θ cm H 3 2 U 1 y H H 3 2 U 1 y H 2 Θ x, y dy 2 dy It can, therefore, be written as Θ cm = m= G m exp 2λ 2 mx* 3 The first three coefficients and eigenvalues are: m G m λ m 2.91 2.83 1.533 32.1 2.153 93.4 Everything we want to know about the temperature profile is in the solution given above. There is a simple one-term approximation since the second eigenvalues is so much larger than the first. Θ cm =.91 exp ( 1.89x*) ChE 333 4

The Local Heat Transfer Coefficient The local heat transfer coefficient is defined in terms of the heat transferred to the fluid in a differential distance along the exchanger. dq H = - UH ρc p dt cm = h ln (T cm - T H ) dx Recalling the non-dimensionalization, we can express the local heat transfer coefficient, h ln, as h ln = UH k UH 2 Θ cm dθ cm dx * From the solution for the dimensionless mixing cup temperature, we obtain 4h ln H k = Nu ln = m= 8 3 λ 2 mg m exp 2λ 2 m x */3 m= ( ) ( ) G m exp 2λ m 2 x*/3 Here the Nusselt number is defined with a length scale 4 H, that is the hydraulic diameter. D H = 4 (cross-sectional are)/(wetted perimeter) 1 The one-term appoximation to Θ n provides the limiting value for the Nusselt number for large x* This is generally valid if x* >.1 Nu ln = 8 3 λ 2 = 7.55 1 Note that D H = 4 2HW 2W + 4H = 4H 1+ 2H W ChE 333 5

We will find shortly that, though the local Nusselt is useful, we will have recourse to an average value over the length of the heat exchanger Nu L 1 * x * L Nu x ln ( x *) dx* L Though it would appear that we would have to return to the detailed solution for Θ cm, the energy balance yields us a simpler form to evaluate. Short Time Solutions Nu L 4 * x ln 1 * L Θ cm x L ( ) The equations describing the entrance region of the Parallel Plate Heat Exchanger are identical save for the the midplane condition Boundary Conditions u(y) T x = α 2 T Dy 2 T = T1 at x = for all y Initial temperature profile T T 1 as y Free stream condition T = TH at y H for all x Wall temperature ChE 333 6

Approximate Velocity Profile In the entrance region, the thermal boundary layer is thin compared to the width of the channel so that the velocity profile can be approximated by a linear profile and the problem is then posed as Boundary Conditions βy T x = α 2 T y 2 T = T1 at x = for all y Initial temperature profile T T 1 as y Free stream condition T = TH at y H for all x Wall temperature Here from an expansion of the velocity profile near the wall, β = 3U/H. The problem is identical in statement to the falling film problem discussed in Mass Transfer and the solution is the same. We can use a combination of variables η = y β 9αx 1/ 3 Thsolution for Θ is Θ = η exp ξ 3 dξ = exp ξ 3 dξ η exp ξ 3 dξ Γ 4 3 ChE 333 7

Nusselt Number Relations The heat flux at the boundary is q y x = k dt dy y= = k T H T 1 β 9αx* 1 /3 The heat transferred per unit width is Q H W = k L dt dy y= dx = 3k T H T 1 2Γ 4 3 β 9α 1/3 so that the local heat transfer coefficient can be expressed as h x = q y = k β 9αx 1 /3 T H T 1 Γ 4 3 The local Nusselt Number relation is Nu(x) = 4h(x)H k = 4 UH2 3αx Γ 4 3 1 /3 = 3.12 x* 1/ 3 Expressed as an average Nusselt Number the relation becomes Nu L = 4h L H k = 2.95 Re Pr H 4L 1/3 ChE 333 8